Presumably, if you put a detector in front of one of those slits, the particles cannot pass through both slits, and thus you wouldn't have those two moving collections of particles. You would have one moving collection, which would simply fan out without interference.
Surely in some worlds you detect a particle and in others you don't, so there are plenty of particles going through each slit on aggregate.
This is the first explanation of a 'many worlds' theory of quantum mechanics that has ever made sense to me. The animations are excellent:
http://www.preposterousuniverse.com/blog/2014/12/16/guest-post-chip-sebens-on-the-many-interacting-worlds-approach-to-quantum-mechanics/