Both a brass rod and a bronze brush are significantly lower on the hardness scale than a steel barrel. This means they should not directly scratch the barrel, if used carefully. However, debris picked up in the bristles and along the shaft of the rod has no guarantees from Herr Mohs. If you're not paying attention and rubbing the sides of the crown, hammering it through, and otherwise applying pressure at the wrong spots at the right angles, it will erode away what you don't want eroded.
Some brushes, even though bronze, might have a steel wire twist core, or threaded base (it could also be soft aluminium). So if this contacts the barrel, particularly while doing the drill twisteroo, you're in trouble.
The drill procedure is usually performed with a steel brush or steel wool wrapped on a bronze one, or a mild abrasive such as JB Bore Paste. And it is usually done to polish a rough chamber which might get shells stuck on extraction, or to remove neglectful amounts of rust or lead build-up. It's easy to go too far!
If you have to do this regularly, there is something wrong with your barrel, i.e., major pitting. This will be holes left behind by rust, after the rust has been cleaned away, and they will collect copper jacket, lead, and powder fouling.
