It depends on the specific terms of the individual library’s contract.
I do not have music in any non-exclusive re-titling libraries. So, I cannot comment on the terms of their contracts.
My music is either in exclusive libraries, where I was paid up front, or in libraries where I own the publishing. I do not shop the music that is in exclusive libraries. I’m free to do what I want with music that I publish and/or own 100%.