A research team from West Chester University in Pennsylvania recently presented a look into one of the more pressing issues of our time: what's up with the doggy paddle?
They found that the gait is identical to what could be described as a dog's natural jog -- not a sprint, but a faster-than-walking movement in which each leg moves in tandem with the leg diagonal to it. The research tested six different breeds of dog, and found the gait to be very similar among all of them.
Efforts to teach the dogs to do other strokes, like the butterfly or breast-stroke, were not conducted. Maybe next time.