I actually think a global layer are separated layers and not one consistent layer that will exist throughout the game (which is indeed not how you might expect them to run), I may be wrong on that one but that would explain it.
Making the sprite global might work but could be problematic as I do not think it is possible to togle the globality parameter inside the event sheet (and also is not something I'd recommand doing for sprites without a lot of care, basically on ending the layout you would have to destroy the ones that still exists and are not on the global layer).