Here is how I clean. I patch with Butch's short stroking about four patches (.22 cal. Sinclair jag, with 1 3/8 sq patch in 6mm bore), with particular attention to the throat, about 20 short strokes there on the first patch. If the barrel is very dirty I just pass the first patch through without short stroking. After the patching, I wet a brush (bronze) with Butch's and do about 20 cycles, being careful to stop on the in stroke, when the brush just clears the muzzle. Then I run another couple of wet patches through to clean out the brush residue, and wait 20 minutes to let the solvent work. After that I run another wet patch and look for color. If there is only a very light tint, I don't worry. I dry the bore, and then the chamber, wipe off the muzzle, lube the locking lugs, and cocking cam with with bolt grease, and I am good to go. Brushes should have some resistance. I buy them from a benchrest supplier by the dozen, and change as needed. They are a "cost of doing business". For neglected barrels, that do not come clean after doing this procedure twice, I wet the bore, and rub JB into a patch, used on a 6mm jag, and short stroke it through, spending more time at the back, and being careful at the muzzle. Sometimes I do this twice, and then clean it all out. If I am cleaning at home, and have the time, I use Wipeout for two hours, and then over night, dry, and then brush with Butch's. This is easier on the barrel, and takes less work. Wash any Wipeout off your hands with soap and hot water, every time you put it in a barrel, or patch it out. You won't feel it on your hands, but if you wipe your face, the edges of your eyelids will start to feel it, even though you have thoroughly wiped your hands.
Added later: Perhaps I should point out that I use a state of the art bore guide, that does a lot more than just keep the solvent out of my action, and that I wipe off my rods every time they come out of the bore. Another thing, I pay a lot of attention to keeping the as coaxial with the bore as I can, and slow my stroke, and have made changes my stance to help accomplish this. Putting a lot of rod strokes in a bore can be dangerous to its well being, if it is done with the wrong equipment and poor technique ( which, unfortunately, is almost universal). In my view this is very important, and I am constantly amazed at how shooters will spend so much on other things, but go cheap and sloppy on cleaning.