The Born rule does not enter into the Everettian interpretation as the consequence of an additional dynamical process. Precisely how it enters is a matter of some controversy. I think the most promising program is the Savage-style decision theoretic approach adopted by the Oxford Everettians, which I discussed here. Whatever you think about the success of this approach, it is an attempt at a rigorous foundation for the Born rule that is missing from most other interpretations. I would also argue that it is a much simpler and more philosophically sound approach than postulating arbitrary discontinuities in physical law.