diff -u b/core/modules/media/src/MediaAccessControlHandler.php b/core/modules/media/src/MediaAccessControlHandler.php
--- b/core/modules/media/src/MediaAccessControlHandler.php
+++ b/core/modules/media/src/MediaAccessControlHandler.php
@@ -39,6 +39,13 @@
         if ($account->hasPermission('edit own ' . $type . ' media') && $is_owner) {
           return AccessResult::allowed()->cachePerPermissions()->cachePerUser()->addCacheableDependency($entity);
         }
+        // @todo: Deprecate this permissions in https://www.drupal.org/project/drupal/issues/2925459
+        if ($account->hasPermission('update any media')) {
+          return AccessResult::allowed()->cachePerPermissions();
+        }
+        if ($account->hasPermission('update media') && $is_owner) {
+          return AccessResult::allowed()->cachePerPermissions()->cachePerUser()->addCacheableDependency($entity);
+        }
         return AccessResult::neutral()->cachePerPermissions();
 
       case 'delete':
@@ -48,6 +55,13 @@
         if ($account->hasPermission('delete own ' . $type . ' media') && $is_owner) {
           return AccessResult::allowed()->cachePerPermissions()->cachePerUser()->addCacheableDependency($entity);
         }
+        // @todo: Deprecate this permissions in https://www.drupal.org/project/drupal/issues/2925459
+        if ($account->hasPermission('delete any media')) {
+          return AccessResult::allowed()->cachePerPermissions();
+        }
+        if ($account->hasPermission('delete media') && $is_owner) {
+          return AccessResult::allowed()->cachePerPermissions()->cachePerUser()->addCacheableDependency($entity);
+        }
         return AccessResult::neutral()->cachePerPermissions();
 
       default: