When I'm doing an extreme clean I clamp my barrels in a leather padded vise. Then I push folded Bounty paper towels through wet with solvent expelling them out the front of the bore into a trash can.
I then remove my rod and wipe it down and repeat until I feel all the loose fowling has been pushed out. (I've saw guys work a dirty patch back and forth...to me it makes little sense to spread around what I'm trying to get out).
I then brush back and forth with a bronze/brass bristle brush keeping it well saturated with solvent going in and coming back...at the point that I feel is enough brushing I go back to pushing a few solvent soaked paper towels through followed by dry ones and followed up with an oiled one. Myself I like Ezzox as a rust preventive.
I'm sure Mr. Sniffle'bean will be along in a minute to explain his old under wear cleaning technique.