Bronze brushes with a brass wire core cannot damage your barrel unless you make a habit of running it partially into the bore and then reversing the direction. Barrels are made of hardened steel and since bronze and brass are much softer metals, they are excellent for brush construction and safe to use regularly.
I recommend Tipton "Best" bore brushes. The brass wire core is looped through the threaded base so it will not separate while using it. They also have more bristles than most others that I have seen which means they get the job done faster. When used properly, they will not harm your barrels in any way. I have had Hoppe's brushes come apart where they are crimped, leaving the brush stuck in the barrel on a couple of occasions. They were easy enough to remove using a jag to push them out.
Also, I have found using a cable and pulling the brushes and patches from the breech to the muzzle can be done on any weapon I have. Most will tell you never to push a tool into the muzzle. When using a rod, on some weapons it is not possible to push from the breech to the muzzle and you risk damaging the crown if you push from muzzle to breech.
Maybe a little more than you were asking for, but I do love my Tipton brushes and jags. Once you use them, everything else seems like cheap junk!