I avoid all metallic brushes as well as the harsh aggressive bore cleaners that use primarily ammonia. I use the Patch Out bore cleaner (and their accelerator if faster cleaning is needed). Simply run one patch through the bore to get the loose major stuff. Then wet the bore with a loose patch or nylon brush. Let it sit overnight (horizontal, but muzzle slightly down from level). The next day just run a wet patch through. Most likely will be the color of new blue jeans. That is the copper it pulled out. If it is heavily fouled, soak it more. The cleaner causes no damage to the bore, and is just as rust resistant as oil.
Brownells or Midway sell patch out as well as rods, jags, patches, brushes.
Best to use jags that are nickel plated, or the solvents will attack the brass jags as well.