I always guide a patch or brush out of the muzzle so the rod doesn't tap even lightly against the lands.
I prefer to catch the brush on the way out with a clean cloth and dab away any accumulated fouling before pullig it back through. If you use the brush only in one direction the bristles will be bent in one direction only and reduce its effectiveness as the bristle takes a set.
I then use a jag in one direction only, several patches from breech to muzzle to push out dislodged solids.
I use an undersized jag and thick felt patches. The felt picks up the solids but the loose fibers hold them without pressing them against the bore.
When using a patch in a slotted head I don't let it leave the muzzle. I push till just visible then draw back carefully. I use a loose patch to swab rather than scrub.
Its spray ,then brush, then jag, then spray and swab , then brush and jag then spray and swab and then let sit for awhile and repeat.
I usually clean as well as I can then spray and swab and let set for several hours and do it all again. Stuff missed the first time can float loose hours later.
PS
I got a can of Gunslick foaming cleaner, I'll give it a try next time I do any shooting.