I've read that this is a bad idea if the rounds could have stick powder in them. Supposedly the vibrations can cause the sticks to break into smaller particles increasing the pressures. I have no idea if this is true but I have avoided it just in case. I have tumbled some blank rounds and they cleaned up pretty well.
I tested the process 5+ years ago. 3 sets of 20 rounds, one not tumbled, 1 tumbled 30 min, 1 tumbled 24 hours. Average Velocity of all 3 was within 10 FPS of each other. 223 in an AR.
I do not allow any type of polishing media or polishing additive to touch any components involved with my match rifles. Polishing media will stay on and in the case, allowing small amounts of abrasive compound into the barrel of your rifle. It just doesn't make sense to allow it in a high dollar barrel. I have no proof it's done any damage to barrels in the past, but it's just a personal choice.