Unlikely in boxing, only because they're trained not to follow the person down
My point about boxing was not that they follow the opponent down but that the opponent in a bout is allowed up time and time again when they shouldn't be. MMA doesn't have a standing or lying count, you go down KO'd the fight is finished, no one continues fighting when they are unable to.
I've seen a boxer in a fight in a pub kick a man in the face he'd just dropped.
When ;overkill' happens in MMA which isn't as often as you'd think given the publicity, it's the refs job to stop it so often it's poor refereeing to blame as much as anything. An MMA fighter who allows himself to get carried away is not a good fighter and will, like an angry fighter, soon lose their fights and be out of the game. Promoters don't want that type, because others don't want to match with them, and they will soon be dropped.