One must distinguish different varieties of MWI. There is an old version of the interpretation (which, I think, is basically what most informed laypeople think of when they hear "MWI") according to which worlds cannot interact. This is because "world-splitting" is a postulate that is added to the Schrodinger dynamics. Whenever a quantum measurement occurs, the entire universe (the ordinary 3+1-dimensional universe we are all familiar with) duplicates (except that the two versions have different outcomes for the measurement). It's basically as mysterious a process as collapse, perhaps even more mysterious.

This is different from the MWI most contemporary proponents accept. This MWI (also called "Everettianism" or "The Theory of the Universal Wavefunction" or...) does not actually have full-fledged separate universes. The fundamental ontology is just a single wavefunction. When macroscopic branches of the wavefunction are sufficiently separate in configuration space, one can loosely describe it as world-splitting. But there is nothing preventing these branches from interfering in principle, just as microscopic branches interfere in the two-slit experiment. There is no magical threshold of branch size/separation where the Schrodinger equation no longer permits interference. And in MWI understood this way, the Schrodinger equation is all the dynamics there are. So yeah, MWI does allow for the interaction of "worlds" in principle. The reason we never see this happening at a macroscopic scale is usually explained by appeal to special initial conditions (just like the thermodynamic arrow of time).

ETA: And in some sense, all the separate worlds are actually interacting all the time, just at a scale that is impossible for our instruments to detect.