Starting from the chamber, push a solvent-saturated patch completely through the bore to drive debris out towards the muzzle in a single pass. Replace the patch with a fresh one, and scrub the bore five or six times before replacing the patch again. Keep going until the patch remains clean. Next, saturate the bore brush in solvent, and work it back and forth through the full length of the bore about six times. Let the solvent sit in the barrel for 15 minutes, then go back to pushing solvent-soaked patches through until they come out clean.
By now, most of the powder residue should be removed, but more work is required to get rid of copper fouling (if copper is present, a good solvent will chemically react with it and leave telltale green or blue residue on the white patches). Bore brushes won’t help here. The only solution is to keep the bore saturated with solvent, and run a patch through it every 10 minutes. Once the patch comes out clean, the copper has been successfully removed.
Finally, push through a fresh patch to dry the bore, then lightly coat the bore with a preservative such G-96 Gun Treatment. Once you are finished, use the bore mop to wipe clean the chamber.