Patches & brushes, both directions. Over the long term I've never seen any damage or deterioration of accuracy from either.Generally speaking, do you scrub your brass brushes, patches both directions, back & forth, back & forth ? or just down the bore, and remove the patch or brush, and start over from the breach?
Are you saying that a nylon brush can damage a stainless steel barrel crown?I never drag anything back across the crown as it’s not necessary and can be damaging. Numerous barrel makers also recommend against doing it due to chance of damaging the crown. With the quality cleaning chemicals available I find myself not working as hard to clean any of my bolt rifles anymore.
With regards to cleaning a carbon ring out I use a .410 or 20 gauge bore mop and let it soak a bit and then work on it. Usually does the trick very quick.