We've had mice in my building. When I started catching them, I would take care to dispose of them in a humane way.
After catching more than 37 of them, the most "humane" behavior I can rustle up is to keep from screaming "DIE, RODENT, DIIIIIIIE!!!!!" as I put the glue trap in a garbage bag.
Holly likes to sniff them out, but she wouldn't touch them with a 10-foot chew toy. She thinks it's up to humans to touch them and dispose of them. But she likes when I get really, really upset at them.