Having never played the game, this is just a guess...
It's probably just lag. For example, at one instant in time, your opponent could be moving toward an edge. They see themselves going toward the edge, and you see them going toward the edge. They're not idiots, so they turn away from the edge and stay on the road. But, your DS does not immediately receive the signal indicating that they turned. They see themselves stay on the road, but your DS still thinks they are going straight towards the edge, so you see them fall off. Then your DS receives the signal of their updated position, and suddenly you see them back on the road again. This can be really annoying in games, but if it's any consolation, if your connection is lagging, their connection might be lagging too.