ok so i'm not sure if this has any relevance whatsoever to actual battling, but when I send in a Pokemon, there are some of them like milotic, samurott or Kyurem there is a sort of cloud thing that poofs out on their side of the battle field, & it looks similar to part of the rock slide animation. It has always made me curious to what this is & it only ever happens to one Pokemon at once, like if I sent in Kyurem against a mitotic, only the kyurem would make the cloud. I think it might be related to the weight or size of the Pokemon but I don't know.
All Pokemon make that smoke once they are sent into battle from a type of Pokeball. So if you enter a battle with a wild Pokemon, your Pokemon will have the smoke when they hit the ground, but since the wild Pokemon doesn't come out of a Pokeball they just slide into the battle. If you enter a battle with another trainer both yours and their Pokemon will have the effect.
For some Pokemon there is also some "yellow dust" that appears, only on the heavier Pokemon. See this question for more on that.