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There's no mechanism to it other than the mechanism that every interpretation of QM already has for describing the evolution of non-macroscopic quantum systems. MWI just says that large systems and small systems aren't separate magisteria with different laws.
"Worlds" and "branching" are epiphenomenal concepts; they're simplifications of what MWI actually talks about (see Decoherence is Pointless).
It doesn't matter whether branching occurs at a point of or at during some blob of time, probabilistic or otherwise, it's a central part of MWI and you need an equation to describe when it happens. And that equation should agree with the Born probabilities up to our observational limits. Likewise for collapse in theories that invoke collapse. Otherwise it's just hand-waving not science.
What is or is not a "branch" is unimportant. If you have read the link you'll know that a "branch" is not a point mass but a blob spread out in configuration space. All MWI needs is "the probability density of finding oneself in point x in the wavefunction is the amplitude squared at that point". It's standard probability theory then to integrate over a "branch" to find your probability of being in that branch. But the only reason to care about "branches" is because the world looks precisely identical to an observer at every point in that branch.