OTIS... is the way to go.....Irecomend the tactical kit, but they have a kit specifically for small calibers as well....
http://www.otisgun.com/

As always from the breech to muzzle......

When you fire your gun, the powder residue and dirt are in the barrel. The chamber and receiver are clean. If you run a brush or patch from the muzzle end you will push this dirt, residue, and moisture into the chamber and receiver. This is a major cause of stuck cases or problems with lever actions and auto-loading rifles and shotguns. If you push or pull a brush back toward the chamber, you will notice the brush will throw the debris from the bore back into the chamber and locking lugs.