Take some Hoppe's #9 on a patch and saturate the carbon areas on the face of the cylinder, the chambers, the bore, around the forcing cone, behind the ejector star, inside the top strap, around the yoke, and the recoil shield behind the cylinder. Anywhere where you see carbon buildup. Let it sit overnight, then go back and do it again the next day. After two days of soaking, use a copper brush and scrub it away. This will remove the fouling without using abrasives. It works.
Stay on top of it by repeating this method after every shooting session and it should be easy to maintain.