Index: src/js/_enqueues/admin/site-health.js
===================================================================
--- src/js/_enqueues/admin/site-health.js	(revision 47908)
+++ src/js/_enqueues/admin/site-health.js	(working copy)
@@ -10,19 +10,34 @@
 
 	var __ = wp.i18n.__,
 		_n = wp.i18n._n,
-		sprintf = wp.i18n.sprintf;
+		sprintf = wp.i18n.sprintf,
+		data,
+		clipboard = new ClipboardJS( '.site-health-copy-buttons .copy-button' ),
+		isDebugTab = $( '.health-check-body.health-check-debug-tab' ).length,
+		pathsSizesSection = $( '#health-check-accordion-block-wp-paths-sizes' ),
+		successTimeout;
 
-	var data;
-	var clipboard = new ClipboardJS( '.site-health-copy-buttons .copy-button' );
-	var isDebugTab = $( '.health-check-body.health-check-debug-tab' ).length;
-	var pathsSizesSection = $( '#health-check-accordion-block-wp-paths-sizes' );
-
 	// Debug information copy section.
 	clipboard.on( 'success', function( e ) {
-		var $wrapper = $( e.trigger ).closest( 'div' );
-		$( '.success', $wrapper ).addClass( 'visible' );
+		var triggerElement = $( e.trigger ),
+			successElement = $( '.success', triggerElement.closest( 'div' ) );
 
-		wp.a11y.speak( __( 'Site information has been added to your clipboard.' ) );
+		// Clear the selection and move focus back to the trigger.
+		e.clearSelection();
+		// Handle ClipboardJS focus bug, see https://github.com/zenorocha/clipboard.js/issues/680
+		triggerElement.focus();
+
+		// Show success visual feedback.
+		clearTimeout( successTimeout );
+		successElement.removeClass( 'hidden' );
+
+		// Hide success visual feedback after 3 seconds since last success.
+		successTimeout = setTimeout( function() {
+			successElement.addClass( 'hidden' );
+		}, 3000 );
+
+		// Handle success audible feedback.
+		wp.a11y.speak( __( 'Site information has been copied to your clipboard.' ) );
 	} );
 
 	// Accordion handling in various areas.
Index: src/wp-admin/css/site-health.css
===================================================================
--- src/wp-admin/css/site-health.css	(revision 47908)
+++ src/wp-admin/css/site-health.css	(working copy)
@@ -203,22 +203,16 @@
 }
 
 .site-health-copy-buttons .copy-button-wrapper {
+	display: inline-flex;
+	align-items: center;
 	margin: 0.5rem 0 1rem;
 }
 
 .site-health-copy-buttons .success {
-	display: none;
 	color: #40860a;
-	line-height: 1.8;
 	margin-left: 0.5rem;
 }
 
-.site-health-copy-buttons .success.visible {
-	display: inline-block;
-	height: 30px;
-	line-height: 2.30769231;
-}
-
 .site-status-has-issues.hide {
 	display: none;
 }
@@ -452,6 +446,10 @@
 		padding-bottom: 0;
 		font-weight: 600;
 	}
+
+	.wp-core-ui .site-health-copy-buttons .copy-button {
+		margin-bottom: 0;
+	}
 }
 
 /* The breakpoint is usually at 960px, the additional space is to allow for the margin. */
Index: src/wp-admin/site-health-info.php
===================================================================
--- src/wp-admin/site-health-info.php	(revision 47908)
+++ src/wp-admin/site-health-info.php	(working copy)
@@ -99,7 +99,7 @@
 			<button type="button" class="button copy-button" data-clipboard-text="<?php echo esc_attr( WP_Debug_Data::format( $info, 'debug' ) ); ?>">
 				<?php _e( 'Copy site info to clipboard' ); ?>
 			</button>
-			<span class="success" aria-hidden="true"><?php _e( 'Copied!' ); ?></span>
+			<span class="success hidden" aria-hidden="true"><?php _e( 'Copied!' ); ?></span>
 		</div>
 	</div>