Larry Niven, in "All the Myriad Ways," pointed out that given all these parallel worlds, not only does everything become possible, but everything possible becomes necessary--if something could happen, then in some world, it did happen.What he doesn't discuss (nor does anyone else that I've seen discussing this) is that such a physics eliminates the possibility of drama because it eliminates the possibility of anything mattering. Imagine that you are reading a riveting story where a hero rescues the damsel just before the train runs over her. Great news, right? Well, not really because in some other universe the hero didn't make it and the damsel got mashed. Why should we care more about the universe where the damsel is saved and ignore the universe where the damsel gets mashed?
And if you start to think about this too much, it becomes unmanageable. You realize that everything happens.
You can take it clear down to the quantum level--in fact, you're required to, if you want to approach it logically. There's no reason that only differences resulting from human action should exist; the rationalistic universe of science fiction does not give humanity special privileges. Any time there is any event that could go more than one way--the breakdown of a specific atomic nucleus, for example--then it must go all the possible ways, each in a different universe. If something has a one-in-a-million chance of happening, then it does happen in an infinite number of universes--and it doesn't in 999,999 times as many.
You might argue that we care about the universe where the damsel is saved because that is the one where the story takes place. But not really. There was just one universe up until the branch point where the damsel either got saved or didn't. Where the universes branched, we just picked the happy one to follow. And I'm simplifying here; in fact there are an infinite number of branch point and an infinite number of damsels smashed. Taken as a whole, this one incident (in the new physics) turns out to be a greater tragedy than all of the tragedies in human history (in the old physics) going by sheer numbers of tragic deaths.
How can there be drama when you know that by hypothesis of the story, every good thing that happens is accompanies by an infinite number of grotesquely evil things that happen? How can you even have moral choices? Sure, the hero can save this damsel, but then he is condemning some other damsel in the possible world where he didn't save her. Why not let his own world be the one where she is not saved, thereby letting some other him in another world live happily ever after and taking the tragic consequences himself? Wouldn't that be the noble action?
In a multiverse where everything that is possible is real, nothing real matters.
You don't have to go to such desperate lengths to have a respectable science-fiction of parallel universes. Instead, suppose that universe splitting is a rare event rather than happening an infinite number of times per second. It might be related to the build up of dark energy over time to a critical point that requires bifurcation to preserve some conservation law. It might happen once every thousand years or once every two or three hours. Make it frequent enough to give you the selection of worlds that you need for your story and infrequent enough that the reader cares what happens to the characters of one particular world.