"Branching," to the extent that branching is a good word for what happens, already is known to be a behavior of wavefunctions: as a Gaussian pulse moves, it spreads out (due to dispersion inherent to the Schrodinger equation), and we as humans can arbitrarily call that branching. (But, like I said, it's continuous instead of discrete like the word branching would imply.)
So, what remains is to explain why we find the universe in a definite state, if it time-evolves into something other than specific eigenvalues. But, first, I'll ask you: what happens if you put a manned capsule inside of the isolated particle box, so that the person inside the capsule starts dispersing too?
0) quantum mechanics is great, but it can lead to quantum systems described as a superposition of states and then we need to explain why we find the universe in a definite state.
1a) one option is to say that the wave function collapses to a definite state.
1b) another option is to say that there is a multiverse... and what remains is to explain why we find the universe in a definite state.
Solution (a) may be ugly, but “solution” (b) sets you back to the starting point!
As for your question: I don't know. Let's say the Schroedinger equation is actually not perfectly linear and there is spontaneous collapse which happens quite rapidly for a system with N~10^30 particles. Now what?
In a nutshell the idea of multiverse is that the entire universe evolves as an isolated system, without any wavefunction collapse.