Actually WV you are a backward on your cleaning. What I do is first run a wet solvent patch through the bore from the chamber using a cleaning rod with a jag (it comes off at the muzzle when you pull the rod back out). Then dip the brush in solvent and run it back and forth through the bore 5 or 6 times. Then dry patch with jag. Again with brush dipped in solvent 5-6 times. Then dry patches, wet patch, dry patches, etc. until patches come out clean. Then a couple drops of oil on patch.
You should not run your patch back and forth through the bore, especially when you first start cleaning. The initial wet patch pushes most of the loose fouling out of the bore and 'pretreats' the bore with solvent. The brush then loosens any fouling adhering to the bore.
Another note, if you have a copper buildup, you may want to run a wet patch of a good copper solvent through the bore and let it sit for about 5 - 10 mins (read the directions on the bottle). Then run a dry patch through. A blue residue on the patch is the oxidized copper coming out. Keep doing this until patches come out clean. Don't forget to oil bore to neutralize the copper solvent.