To clarify, would the RP be jumping to drastically different places or different versions of our universe? Most of the time when I hear multiverse I think of a universal setup where you can leap between the consequences of two slightly different choices and the resulting slightly different universes(think steins;gate) but I'm not sure if this is the kind of vision you had in mind. 

For whatever reason, I use 'falls into another world' as a plot crutch pretty often, so I may be open.