A lot of times I will run a dry brush down the bore to get most of the residue out of the barrel. Just 2 or 3 strokes and tap the muzzle lightly on my 1" thick rubber floor mat. Next I put the barrel (breach end) into hot soapy water. Dip the brush in the water and scrub the bore. Followed by patches to flush and mop the bore with the soapy water.
So my brush does the majority of the residue removal and the patches do the finish wiping and drying.