Taro Clockwork Post

Private is the default post status but if you want another case, use filter hook for that.

// Filter status if post type is `product`
add_filter( 'tscp_expired_status', function( $status, $post ) {
   if ( 'product' == $post->post_type ) {
       $status = 'no-stock';
   }
   return $status;
}, 10, 2 );

If status is false, this plugin doesn’t change post status.
In such situation, you might need adding any custom field to post.
Use another action which will occur just after tscp_expired_status.

// do something just after post status is/isn't changed.
add_action( 'tscp_post_expired', function( $post ) {
   // Post is still publish,
   // But add some custom fields
   update_post_meta( $post->ID, '_not_in_front_page', true );
} );

If you are low-resource environment, you might need low frequency.
For example, assume that your site allows posts to be expired within 10 min.
Use hook to delay interval.

add_filter( 'tscp_cron_interval', function() {
  // Change interval from 60 sec to 600 sec.
  return 600;
} );