Some people believe in reincarnation. I've proposed the idea that being a verser could possibly be an afterlife. Well, it is, but anyway. What if the two were combined? You are reincarnated as a different person throughout the entire Multiverse? I'd love to hear what Osevens has to say.
Generally, theories of reincarnation assume that the spirit or soul is brought to a new body through birth. That's rather different from versing (although there is the possibility of the Zygote Experience).
It would probably be noticeable if everyone who died in any universe suddenly appeared in a different universe as a fully grown person. I think we can rule out versing as the normal afterlife, unless we assume that this universe is protected against it and other universes consider the appearance of millions of fully-grown people from nowhere normal.
Most religions that teach reincarnation also teach transmigration of souls--you don't have a human soul but a creature soul, and if you're good you'll come back next time as a higher being such as a cow, and if you're bad you'll return as a lower creature such as a worm. Of course, you can divorce the two concepts (and most moderns who believe in reincarnation do so, all previous lives being human lives (and, oddly, usually the lives of wealthy or influential people)). You just need to consider what justification you have for making that decision.
There are games and works of fiction in which characters become someone else, but they're not generally well known. In fact, I only know of them because people who were working on them contacted me about them. To me, they didn't seem to work as well: suddenly I awaken, and I'm a fully-trained Roman Gladiator preparing to fight in the arena, and look at this I suddenly have all these combat skills I don't remember learning; and now I'm elsewhere, and I'm the third princess of a medieval kingdom, and boy, would those now-forgotten battle skills have come in handy when the castle is sacked.
--M. J. Young
MJ's answer +1