I'm not an experienced boxer, but here's my 2 cents.

Punching power comes mainly from 3 things: Form, Muscular Strength, and Body Weight. Big people hit hard. Strong people hit hard. And people that know how to throw a punch properly hit hard. Improve any of these 3 things and in theory, you should be hitting harder.

With that said, punching while standing up with light weight dumbells in your hands probably won't make you hit much harder. The dumbells are weighing your fists down so if anything, it will help you keep your hands up when your hands would otherwise feel heavy after a long round. Has anyone here tried "punching" with dumbells while laying on a bench? I guess it would be very similar to a dumbell benchpress, but the form would be slightly different. I would imagine that if someone who was roughly my weight and my skill level, could benchpress twice as much as me, he could probably hit a lot harder than me (everthing else being equal).

The muscles used when punching depend on the punch but the most active muscles are probably in the chest, shoulders and triceps. But if you're punching with proper form, nearly every muscle in your body is tied in. Imagine that your legs were made of jello. Sounds crazy, but if that were the case, throwing a powerful punch at your opponent would knock you over backwards. Or imagine if your forearm and wrist muscles were too weak to keep your fist firm on impact. You might as well be hitting with him with a pillow. Or if your abdominals were too weak to turn your body into a right cross... virtually every muscle in the body is involved in a proper punch.