Most of my flock are RIRs, and I've never had any bullies out of them. Except the roosters, but that's a different story (but I have had nice roosters too!!).
What is their feeder situation like? Chicks are curious little anklebiters (sometimes literally) and are always trying to eat things they shouldn't. If you haven't already, I would sprinkle some feed on the floor of the brooder and see if that helps.