Unpublish / Update Feed Items Automatically

I’m using this plugin to pull in feeds of a custom post type from another WordPress site. We’re using it to pull in “Jobs” to post job listings for that specific location on that location’s website.

Is there a way, for when the plugin crawls the feed, to have it automatically update posts / unpublish posts that are no longer published? For example, several jobs have been unpublished, because the position was filled, but the plugin didn’t remove the feed item. We’ve had other jobs that had updates made, so the published date was updated, but it kept the original date from December.