diff --git a/python/core/auto_generated/layout/qgslayoutitemmapitem.sip.in b/python/core/auto_generated/layout/qgslayoutitemmapitem.sip.in
index 3c3ceff065f..a53557b8439 100644
--- a/python/core/auto_generated/layout/qgslayoutitemmapitem.sip.in
+++ b/python/core/auto_generated/layout/qgslayoutitemmapitem.sip.in
@@ -185,6 +185,13 @@ Returns ``True`` if the visitor should continue visiting other objects, or ``Fal
 should be canceled.
 
 .. versionadded:: 3.10
+%End
+
+    virtual QgsMapLayer *mapLayer();
+%Docstring
+Returns the internal map layer used by this item, if available.
+
+.. versionadded:: 3.10.1
 %End
 
     virtual QgsExpressionContext createExpressionContext() const;
diff --git a/python/core/auto_generated/layout/qgslayoutitemmapoverview.sip.in b/python/core/auto_generated/layout/qgslayoutitemmapoverview.sip.in
index b1f3b684c3d..95e4ec13eb6 100644
--- a/python/core/auto_generated/layout/qgslayoutitemmapoverview.sip.in
+++ b/python/core/auto_generated/layout/qgslayoutitemmapoverview.sip.in
@@ -238,6 +238,9 @@ Ownership of the layer remain with the overview item.
 .. versionadded:: 3.6
 %End
 
+    virtual QgsMapLayer *mapLayer();
+
+
     virtual bool accept( QgsStyleEntityVisitorInterface *visitor ) const;
 
 
diff --git a/src/core/layout/qgslayoutexporter.cpp b/src/core/layout/qgslayoutexporter.cpp
index 96a1aaaf59d..c55c8e2ce03 100644
--- a/src/core/layout/qgslayoutexporter.cpp
+++ b/src/core/layout/qgslayoutexporter.cpp
@@ -1599,7 +1599,7 @@ bool QgsLayoutExporter::georeferenceOutputPrivate( const QString &file, QgsLayou
   return true;
 }
 
-QString nameForLayerWithItems( const QList< QGraphicsItem * > items, unsigned int layerId )
+QString nameForLayerWithItems( const QList< QGraphicsItem * > &items, unsigned int layerId )
 {
   if ( items.count() == 1 )
   {
diff --git a/src/core/layout/qgslayoutitemmap.cpp b/src/core/layout/qgslayoutitemmap.cpp
index 5e334c9b001..1f2b8428e49 100644
--- a/src/core/layout/qgslayoutitemmap.cpp
+++ b/src/core/layout/qgslayoutitemmap.cpp
@@ -1059,10 +1059,14 @@ bool QgsLayoutItemMap::nextExportPart()
       FALLTHROUGH
 
     case Grid:
-      if ( mOverviewStack->hasEnabledItems() )
+      for ( int i = 0; i < mOverviewStack->size(); ++i )
       {
-        mCurrentExportPart = OverviewMapExtent;
-        return true;
+        QgsLayoutItemMapItem *item = mOverviewStack->item( i );
+        if ( item->enabled() && item->stackingPosition() == QgsLayoutItemMapItem::StackAboveMapLabels )
+        {
+          mCurrentExportPart = OverviewMapExtent;
+          return true;
+        }
       }
       FALLTHROUGH
 
@@ -1132,6 +1136,25 @@ QgsLayoutItem::ExportLayerDetail QgsLayoutItemMap::exportLayerDetails() const
               else
                 detail.name = QStringLiteral( "%1: %2" ).arg( displayName(), layer->name() );
             }
+            else
+            {
+              // might be an item based layer
+              const QList<QgsLayoutItemMapOverview *> res = mOverviewStack->asList();
+              for ( QgsLayoutItemMapOverview  *item : res )
+              {
+                if ( !item || !item->enabled() || item->stackingPosition() == QgsLayoutItemMapItem::StackAboveMapLabels )
+                  continue;
+
+                if ( item->mapLayer() && detail.mapLayerId == item->mapLayer()->id() )
+                {
+                  if ( !detail.mapTheme.isEmpty() )
+                    detail.name = QStringLiteral( "%1 (%2): %3" ).arg( displayName(), detail.mapTheme, item->mapLayer()->name() );
+                  else
+                    detail.name = QStringLiteral( "%1: %2" ).arg( displayName(), item->mapLayer()->name() );
+                  break;
+                }
+              }
+            }
             return detail;
           }
 
@@ -2493,8 +2516,10 @@ QgsRectangle QgsLayoutItemMap::computeAtlasRectangle()
 
 void QgsLayoutItemMap::createStagedRenderJob( const QgsRectangle &extent, const QSizeF size, double dpi )
 {
-  // TODO - overview?
-  mStagedRendererJob = qgis::make_unique< QgsMapRendererStagedRenderJob >( mapSettings( extent, size, dpi, true ),
+  QgsMapSettings settings = mapSettings( extent, size, dpi, true );
+  settings.setLayers( mOverviewStack->modifyMapLayerList( settings.layers() ) );
+
+  mStagedRendererJob = qgis::make_unique< QgsMapRendererStagedRenderJob >( settings,
                        mLayout && mLayout->renderContext().flags() & QgsLayoutRenderContext::FlagRenderLabelsByMapLayer
                        ? QgsMapRendererStagedRenderJob::RenderLabelsByMapLayer
                        : QgsMapRendererStagedRenderJob::Flags( nullptr ) );
diff --git a/src/core/layout/qgslayoutitemmapitem.cpp b/src/core/layout/qgslayoutitemmapitem.cpp
index b8f4c119629..40747892844 100644
--- a/src/core/layout/qgslayoutitemmapitem.cpp
+++ b/src/core/layout/qgslayoutitemmapitem.cpp
@@ -134,6 +134,11 @@ bool QgsLayoutItemMapItem::accept( QgsStyleEntityVisitorInterface * ) const
   return true;
 }
 
+QgsMapLayer *QgsLayoutItemMapItem::mapLayer()
+{
+  return nullptr;
+}
+
 //
 // QgsLayoutItemMapItemStack
 //
diff --git a/src/core/layout/qgslayoutitemmapitem.h b/src/core/layout/qgslayoutitemmapitem.h
index 2ffb4a72adf..df3ee92c008 100644
--- a/src/core/layout/qgslayoutitemmapitem.h
+++ b/src/core/layout/qgslayoutitemmapitem.h
@@ -189,6 +189,13 @@ class CORE_EXPORT QgsLayoutItemMapItem : public QgsLayoutObject
      */
     virtual bool accept( QgsStyleEntityVisitorInterface *visitor ) const;
 
+    /**
+     * Returns the internal map layer used by this item, if available.
+     *
+     * \since QGIS 3.10.1
+     */
+    virtual QgsMapLayer *mapLayer();
+
     QgsExpressionContext createExpressionContext() const override;
 
   protected:
diff --git a/src/core/layout/qgslayoutitemmapoverview.cpp b/src/core/layout/qgslayoutitemmapoverview.cpp
index d3f40f6ab6f..8e5283ec653 100644
--- a/src/core/layout/qgslayoutitemmapoverview.cpp
+++ b/src/core/layout/qgslayoutitemmapoverview.cpp
@@ -34,7 +34,7 @@
 
 QgsLayoutItemMapOverview::QgsLayoutItemMapOverview( const QString &name, QgsLayoutItemMap *map )
   : QgsLayoutItemMapItem( name, map )
-  , mExtentLayer( qgis::make_unique< QgsVectorLayer >( QStringLiteral( "Polygon?crs=EPSG:4326" ), QStringLiteral( "overview" ), QStringLiteral( "memory" ), QgsVectorLayer::LayerOptions( map && map->layout() && map->layout()->project() ? map->layout()->project()->transformContext() : QgsCoordinateTransformContext() ) ) )
+  , mExtentLayer( qgis::make_unique< QgsVectorLayer >( QStringLiteral( "Polygon?crs=EPSG:4326" ), tr( "Overview" ), QStringLiteral( "memory" ), QgsVectorLayer::LayerOptions( map && map->layout() && map->layout()->project() ? map->layout()->project()->transformContext() : QgsCoordinateTransformContext() ) ) )
 {
   createDefaultFrameSymbol();
 }
@@ -309,6 +309,11 @@ QgsVectorLayer *QgsLayoutItemMapOverview::asMapLayer()
   return mExtentLayer.get();
 }
 
+QgsMapLayer *QgsLayoutItemMapOverview::mapLayer()
+{
+  return mExtentLayer.get();
+}
+
 bool QgsLayoutItemMapOverview::accept( QgsStyleEntityVisitorInterface *visitor ) const
 {
   if ( mFrameSymbol )
diff --git a/src/core/layout/qgslayoutitemmapoverview.h b/src/core/layout/qgslayoutitemmapoverview.h
index 4d1936c6de0..21edef16037 100644
--- a/src/core/layout/qgslayoutitemmapoverview.h
+++ b/src/core/layout/qgslayoutitemmapoverview.h
@@ -230,6 +230,8 @@ class CORE_EXPORT QgsLayoutItemMapOverview : public QgsLayoutItemMapItem
      */
     QgsVectorLayer *asMapLayer();
 
+    QgsMapLayer *mapLayer() override;
+
     bool accept( QgsStyleEntityVisitorInterface *visitor ) const override;
 
   public slots:
diff --git a/tests/src/core/testqgslayoutmap.cpp b/tests/src/core/testqgslayoutmap.cpp
index 11f6d24b512..c46ded1587d 100644
--- a/tests/src/core/testqgslayoutmap.cpp
+++ b/tests/src/core/testqgslayoutmap.cpp
@@ -1065,6 +1065,7 @@ void TestQgsLayoutMap::testLayeredExport()
 
   map->setBackgroundEnabled( false );
   map->overview()->setEnabled( true );
+  map->overview()->setStackingPosition( QgsLayoutItemMapItem::StackAboveMapLabels );
   map->startLayeredExport();
   QVERIFY( map->nextExportPart() );
   map->createStagedRenderJob( map->extent(), QSize( 512, 512 ), 72 );
@@ -1117,6 +1118,67 @@ void TestQgsLayoutMap::testLayeredExport()
   QVERIFY( !map->nextExportPart() );
   map->stopLayeredExport();
 
+  map->overview()->setStackingPosition( QgsLayoutItemMapItem::StackBelowMapLabels );
+  QgsLayoutItemMap *map2 = new QgsLayoutItemMap( &l );
+  map2->attemptSetSceneRect( QRectF( 20, 20, 200, 100 ) );
+  map2->setFrameEnabled( false );
+  map2->setBackgroundEnabled( false );
+  map2->setCrs( linesLayer->crs() );
+  map2->zoomToExtent( linesLayer->extent() );
+  map2->setLayers( QList<QgsMapLayer *>() << linesLayer );
+  l.addLayoutItem( map2 );
+  map->overview()->setLinkedMap( map2 );
+  map->startLayeredExport();
+  QVERIFY( map->nextExportPart() );
+  map->createStagedRenderJob( map->extent(), QSize( 512, 512 ), 72 );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: lines" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, linesLayer->id() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: Overview" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, map->overview()->mapLayer()->id() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  // labels
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: Labels" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: Grids" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: Frame" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+
+  map->overview()->setStackingPosition( QgsLayoutItemMapItem::StackAboveMapLabels );
+
   // add second layer
   map->setLayers( QList<QgsMapLayer *>() << linesLayer << pointsLayer );
   map->startLayeredExport();
@@ -1390,6 +1452,147 @@ void TestQgsLayoutMap::testLayeredExport()
   QVERIFY( !map->nextExportPart() );
   map->stopLayeredExport();
 
+  map->overview()->setStackingPosition( QgsLayoutItemMapItem::StackBelowMapLabels );
+  map->startLayeredExport();
+  QVERIFY( map->nextExportPart() );
+  map->createStagedRenderJob( map->extent(), QSize( 512, 512 ), 72 );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: Background" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset2): lines" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, linesLayer->id() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset2" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QCOMPARE( map->themeToRender( QgsExpressionContext() ), QStringLiteral( "test preset2" ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset2): points" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, pointsLayer->id() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset2" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QCOMPARE( map->themeToRender( QgsExpressionContext() ), QStringLiteral( "test preset2" ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset2): Overview" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, map->overview()->mapLayer()->id() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset2" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  // labels
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset2): Labels" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset2" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QCOMPARE( map->themeToRender( QgsExpressionContext() ), QStringLiteral( "test preset2" ) );
+  QVERIFY( map->nextExportPart() );
+  // "test preset"
+  map->createStagedRenderJob( map->extent(), QSize( 512, 512 ), 72 );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset): lines" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, linesLayer->id() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QCOMPARE( map->themeToRender( QgsExpressionContext() ), QStringLiteral( "test preset" ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset): Overview" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, map->overview()->mapLayer()->id() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  // labels
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset): Labels" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QCOMPARE( map->themeToRender( QgsExpressionContext() ), QStringLiteral( "test preset" ) );
+  QVERIFY( map->nextExportPart() );
+  map->createStagedRenderJob( map->extent(), QSize( 512, 512 ), 72 );
+  // "test preset 3"
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset3): points" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, pointsLayer->id() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset3" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QCOMPARE( map->themeToRender( QgsExpressionContext() ), QStringLiteral( "test preset3" ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset3): Overview" ) );
+  QCOMPARE( map->exportLayerDetails().mapLayerId, map->overview()->mapLayer()->id() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset3" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->nextExportPart() );
+  // labels
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1 (test preset3): Labels" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QCOMPARE( map->exportLayerDetails().mapTheme, QStringLiteral( "test preset3" ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QCOMPARE( map->themeToRender( QgsExpressionContext() ), QStringLiteral( "test preset3" ) );
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: Grids" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( map->themeToRender( QgsExpressionContext() ).isEmpty() );
+
+  QVERIFY( map->nextExportPart() );
+  QCOMPARE( map->exportLayerDetails().name, QStringLiteral( "Map 1: Frame" ) );
+  QVERIFY( map->exportLayerDetails().mapLayerId.isEmpty() );
+  QVERIFY( map->exportLayerDetails().mapTheme.isEmpty() );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Grid ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::OverviewMapExtent ) );
+  QVERIFY( map->shouldDrawPart( QgsLayoutItemMap::Frame ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Background ) );
+  QVERIFY( !map->shouldDrawPart( QgsLayoutItemMap::Layer ) );
+  QVERIFY( !map->nextExportPart() );
+  map->stopLayeredExport();
+
+  map->overview()->setStackingPosition( QgsLayoutItemMapItem::StackAboveMapLabels );
+
   // but if map is already set to a particular map theme, it DOESN'T follow the theme iteration
   map->setFollowVisibilityPreset( true );
   map->setFollowVisibilityPresetName( QStringLiteral( "test preset" ) );