Index: src/js/_enqueues/admin/privacy-tools.js
===================================================================
--- src/js/_enqueues/admin/privacy-tools.js	(revision 47388)
+++ src/js/_enqueues/admin/privacy-tools.js	(working copy)
@@ -60,6 +60,7 @@
 			$action        = $this.parents( '.export-personal-data' ),
 			$requestRow    = $this.parents( 'tr' ),
 			$progress      = $requestRow.find( '.export-progress' ),
+			$rowActions    = $this.parents( '.row-actions' ),
 			requestID      = $action.data( 'request-id' ),
 			nonce          = $action.data( 'nonce' ),
 			exportersCount = $action.data( 'exporters-count' ),
@@ -68,6 +69,8 @@
 		event.preventDefault();
 		event.stopPropagation();
 
+		$rowActions.addClass( 'processing' );
+
 		$action.blur();
 		clearResultsAfterRow( $requestRow );
 		setExportProgress( 0 );
@@ -78,7 +81,6 @@
 			setActionState( $action, 'export-personal-data-success' );
 
 			appendResultsAfterRow( $requestRow, 'notice-success', summaryMessage, [] );
-			$this.hide();
 
 			if ( 'undefined' !== typeof zipUrl ) {
 				window.location = zipUrl;
@@ -85,6 +87,8 @@
 			} else if ( ! sendAsEmail ) {
 				onExportFailure( strings.noExportFile );
 			}
+
+			setTimeout( function(){ $rowActions.removeClass( 'processing' ); }, 500 );
 		}
 
 		function onExportFailure( errorMessage ) {
@@ -92,6 +96,8 @@
 			if ( errorMessage ) {
 				appendResultsAfterRow( $requestRow, 'notice-error', strings.exportError, [ errorMessage ] );
 			}
+
+			setTimeout( function(){ $rowActions.removeClass( 'processing' ); }, 500 );
 		}
 
 		function setExportProgress( exporterIndex ) {
@@ -118,9 +124,8 @@
 				var responseData = response.data;
 
 				if ( ! response.success ) {
-
 					// e.g. invalid request ID.
-					onExportFailure( response.data );
+					setTimeout( function(){ onExportFailure( response.data ); }, 500 );
 					return;
 				}
 
@@ -127,17 +132,16 @@
 				if ( ! responseData.done ) {
 					setTimeout( doNextExport( exporterIndex, pageIndex + 1 ) );
 				} else {
+					setExportProgress( exporterIndex );
 					if ( exporterIndex < exportersCount ) {
-						setExportProgress( exporterIndex );
 						setTimeout( doNextExport( exporterIndex + 1, 1 ) );
 					} else {
-						onExportDoneSuccess( responseData.url );
+						setTimeout( function(){ onExportDoneSuccess( responseData.url ); }, 500 );
 					}
 				}
 			}).fail( function( jqxhr, textStatus, error ) {
-
 				// e.g. Nonce failure.
-				onExportFailure( error );
+				setTimeout( function(){ onExportFailure( error ); }, 500 );
 			});
 		}
 
@@ -151,6 +155,7 @@
 			$action       = $this.parents( '.remove-personal-data' ),
 			$requestRow   = $this.parents( 'tr' ),
 			$progress     = $requestRow.find( '.erasure-progress' ),
+			$rowActions    = $this.parents( '.row-actions' ),
 			requestID     = $action.data( 'request-id' ),
 			nonce         = $action.data( 'nonce' ),
 			erasersCount  = $action.data( 'erasers-count' ),
@@ -158,8 +163,11 @@
 			hasRetained   = false,
 			messages      = [];
 
+		event.preventDefault();
 		event.stopPropagation();
 
+		$rowActions.addClass( 'processing' );
+
 		$action.blur();
 		clearResultsAfterRow( $requestRow );
 		setErasureProgress( 0 );
@@ -186,12 +194,15 @@
 				}
 			}
 			appendResultsAfterRow( $requestRow, classes, summaryMessage, messages );
-			$this.hide();
+
+			setTimeout( function(){ $rowActions.removeClass( 'processing' ); }, 500 );
 		}
 
 		function onErasureFailure() {
 			setActionState( $action, 'remove-personal-data-failed' );
 			appendResultsAfterRow( $requestRow, 'notice-error', strings.removalError, [] );
+
+			setTimeout( function(){ $rowActions.removeClass( 'processing' ); }, 500 );
 		}
 
 		function setErasureProgress( eraserIndex ) {
@@ -215,7 +226,7 @@
 				var responseData = response.data;
 
 				if ( ! response.success ) {
-					onErasureFailure();
+					setTimeout( function(){ onErasureFailure(); }, 500 );
 					return;
 				}
 				if ( responseData.items_removed ) {
@@ -230,15 +241,15 @@
 				if ( ! responseData.done ) {
 					setTimeout( doNextErasure( eraserIndex, pageIndex + 1 ) );
 				} else {
+					setErasureProgress( eraserIndex );
 					if ( eraserIndex < erasersCount ) {
-						setErasureProgress( eraserIndex );
 						setTimeout( doNextErasure( eraserIndex + 1, 1 ) );
 					} else {
-						onErasureDoneSuccess();
+						setTimeout( function(){ onErasureDoneSuccess(); }, 500 );
 					}
 				}
 			}).fail( function() {
-				onErasureFailure();
+				setTimeout( function(){ onErasureFailure(); }, 500 );
 			});
 		}
 
Index: src/wp-admin/css/forms.css
===================================================================
--- src/wp-admin/css/forms.css	(revision 47388)
+++ src/wp-admin/css/forms.css	(working copy)
@@ -1206,6 +1206,10 @@
 	color: #72777c;
 }
 
+.privacy_requests .row-actions.processing {
+	position: static;
+}
+
 .privacy_requests tbody .has-request-results th {
 	box-shadow: none;
 }