Index: src/wp-includes/default-filters.php
===================================================================
--- src/wp-includes/default-filters.php	(revision 41178)
+++ src/wp-includes/default-filters.php	(working copy)
@@ -391,6 +391,9 @@
 add_action( 'rest_api_init', 'create_initial_rest_routes', 99 );
 add_action( 'parse_request', 'rest_api_loaded' );
 
+// Backcompat filter for mapping `customize_changeset` capabilities.
+add_filter( 'map_meta_cap', 'wp_map_customize_changeset_caps', 0, 4 );
+
 /**
  * Filters formerly mixed into wp-includes
  */
Index: src/wp-includes/post.php
===================================================================
--- src/wp-includes/post.php	(revision 41178)
+++ src/wp-includes/post.php	(working copy)
@@ -168,7 +168,7 @@
 		),
 		'public' => false,
 		'_builtin' => true, /* internal use only. don't use this when registering your own post type. */
-		'map_meta_cap' => true,
+		'map_meta_cap' => false,
 		'hierarchical' => false,
 		'rewrite' => false,
 		'query_var' => false,
@@ -177,21 +177,13 @@
 		'supports' => array( 'title', 'author' ),
 		'capability_type' => 'customize_changeset',
 		'capabilities' => array(
-			'create_posts' => 'customize',
-			'delete_others_posts' => 'customize',
-			'delete_post' => 'customize',
-			'delete_posts' => 'customize',
-			'delete_private_posts' => 'customize',
-			'delete_published_posts' => 'customize',
-			'edit_others_posts' => 'customize',
-			'edit_post' => 'customize',
-			'edit_posts' => 'customize',
-			'edit_private_posts' => 'customize',
-			'edit_published_posts' => 'do_not_allow',
-			'publish_posts' => 'customize',
+			'delete_others_posts' => 'delete_others_customize_changesets',
+			'delete_posts' => 'delete_customize_changesets',
+			'delete_private_posts' => 'delete_private_customize_changesets',
+			'delete_published_posts' => 'delete_published_customize_changesets',
+			'edit_private_posts' => 'edit_private_customize_changesets',
+			'edit_published_posts' => 'edit_published_customize_changesets',
 			'read' => 'read',
-			'read_post' => 'customize',
-			'read_private_posts' => 'customize',
 		),
 	) );
 
Index: src/wp-includes/theme.php
===================================================================
--- src/wp-includes/theme.php	(revision 41178)
+++ src/wp-includes/theme.php	(working copy)
@@ -3121,3 +3121,23 @@
 		clean_post_cache( $post_id );
 	}
 }
+
+function wp_map_customize_changeset_caps( $caps, $cap, $user_id, $args ) {
+	if ( in_array(
+		$cap,
+		array(
+			'delete_customize_changeset',
+			'edit_customize_changeset',
+			'read_customize_changeset',
+			'edit_customize_changesets',
+			'edit_others_customize_changesets',
+			'publish_customize_changesets',
+			'read_private_customize_changesets',
+		),
+		true
+	) ) {
+		$caps = call_user_func_array( 'map_meta_cap', array_merge( array( 'customize', $user_id ), $args ) );
+	}
+
+	return $caps;
+}