"Library" in this context means the papers/books that you have in program X (Zotero, Mendeley, etc.) forming the literary core of a project or topic.
Is there an already-established method to automatically get the latest citations (from papers in your library) or new publications (from authors in your library)? I picture something like having a "Latest developments" panel in the program X showing a list of the newest items which could be of interest according to your library.
I have never seen or heard of something like this and it might well just not exist. However I can see how this could be something very valuable for researchers looking to optimize how they keep up with literature, so maybe there is an app already out there which I might have missed so far.