Hi John,
It's hard to tell you the answer without really knowing what Feynman was talking about in that particular case. I haven't read much of his thoughts on multiple universes.
One possibility is that he was talking about the many worlds interpretation of quantum mechanics. In that case, particles don't interfere with each other in different universes. The particle still interferes with itself in this universe, but every time you try to predict exactly where it will be when it strikes the screen, quantum mechanics tells us that the best you can do to predict the probability of seeing it at different points. The Copenhagen model, which is the most widely accepted, says that the photon is somehow smeared out over all possible values, but "chooses" to only be in one place when it strikes the screen. The many worlds interpretation instead says that the photon doesn't choose only one place, but that it chooses every place, but only one in each universe, so that there are infinitely many universes that appear when it strikes the screen, each of which corresponds to only one value of it's position on the screen.
In either case, the interference needs to happen in only one universe--there is no way for anything to communicate between universes. It's the measurement process that happens after the interference that creates multiple universes.