I'd assume that the number of transferring people would be pretty tiny.
Also there's always the potential of the next universe not sustaining the laws allowing some people to exist from last iteration. I'd assume that eventually the mechanics change too much and certain types just stop working within the confines of the new existence.
Why would the transfer population be tiny? What's the reasoning behind that? Sure some of the population will be sick of life, but I'd presume that as reproduction continues on, there'd be plenty of ones that haven't lived for that long.
Also, chances are the, the universal constants alone would have so much room to differ that nothing will survive any transition.
Or the Kitsune need to adapt to the new universe somehow.
Such as through merging with newverse souls. Odds are is eventually a universe would show up that prevents that from happening.
How? Merging with the new souls requires their stuff to work. But they won't need to unless their stuff doesn't work anymore. And they'd be all dead before they can as a result.
The transfer population would likely be tiny due to the sheer power requirement for even attempting to transfer. I doubt a minor deity could manage it. Ninetail+ Kitsune could probably manage, but at that point they're deific in power, although the powerdrain would be severe enough to put them in the same boat as transferred deities. Principle could definitely manage it, no idea how his energy functions. Eldritch creatures that spawned from The Void could manage it at any power level due to not technically existing. Major deities could probably manage it, but they'd be in the same boat as new deities in the new universe, lack of followers to fuel themselves. Probably similarly powered creatures could manage it.
As for why they would merge with newverse souls, I would assume mechanical deviations aren't so severe as to disable transfers from the prior verse, but may disable transfers from multiple iterations prior. Merging would align them with the most recent iteration and make it safer for offspring transferring to the next verse at the end of the newverse.
This debate is fun by the way. Giving me ideas for a game once I finish Unraveling.