I used to use the 0000 steel wool. Would pull off a piece from the pad and wrap around an appropriate size bore brush screwed into a cleaning rod that had its handle removed. Wound enough on the brush to give a snug fit in the bore. Chucked the rod in a electric hand drill, applied some Hoppe's bore cleaner to the wool and spun it with the drill while moving the rod up and down the barrel. I found that in one barrel it was best to keep the rod moving up and down rather than spinning in one spot and to remove the whole from the barrel while still spinning. The reason being that in that barrel the wool appeared to very slightly cut or scratch the barrel and I was trying to avoid leaving a defined pattern of scratching and leave a polished result. Guess that barrel was very soft steel! Probably all in my mind but that's what I did. Sure cleaned and brightened things up! Finally discovered a bronze wool made by Rhodes American in "fine" and that is what I how use instead of the steel wool. No discernible scratching and works a treat! By the way, you do not need to use new wool, either steel or bronze, each time. You will be able to clean several barrels before you decide enough is enough and change the wool. Hope my experiences help a bit.
J.B.