Ember for Mac, a media manager and annotation tool for screenshots, web archives, GIFs, and other media you don’t want in your normal photo library, can automatically import media from one or more folders you specify.
Just go to Ember > Preferences > Auto-Import to specify folders and even rules based on media file name or type. Once Ember imports media, your rules can optionally move the original file to the trash.
Ember for iOS is an equally powerful standalone app or companion for the Mac version. It can automatically detect and import iOS screenshots in your Camera Roll, though due to iOS limitations it can’t delete the originals.