Here is what I do...take it for what it's worth...which, you didn't pay for it.
First, I use bore guides unless I am HAVING to clean in the field and then I use the Otis stuff.
Second, I don't use copper or brass brushes, just the nylon and I like the aluminum with the nylon.
Third, ALWAYS breech to muzzle.
Fourth, a patch, brush or whatever gets removed before returning to the breech. BREECH TO MUZZLE ALWAYS.
Fifth....did I mention breech to muzzle?
Think of it this way with a new barrel...there are machine marks that need polished out. You are polishing those out with copper jacketed high speed polishing media.
First patch is a wet patch wrapped jag of Hoppes #9. I follow that with patches wrapped on a brush of Naptha (zippo lighter fluid is mostly Naptha, I'm a FUDD and old habits die hard) that I get at Home Depot. I patch it until the patches come out pretty clean. The I run another patch of Hoppes #9. This time, I let it soak for 10 minutes, then I do the Naptha patches again. Follow this with ANOTHER Hoppes patch and more Naptha. I do this until the second Naptha patch comes out pretty clean. Now, here is the copper part. I use Butch's Bore Shine. You gotta be careful with most copper removers. I run a jagged patch of Butch's, followed immediately with another jagged patch of Butch's. I set a timer for 12 minutes (read your directions). Then I start with the patches on a brush with Naptha. Your first Naptha patch is your "reader"...look for green, blue or some similar that indicates that your barrel is rough enough to get some copper fouling. Continue the Naptha until the patches are fairly clean. Then, again with a jagged patch of Butch's followed immediately with another. Set the timer. Etc.
I could give a shit if this uses 100 patches or if it takes me all day.
Once I am convinced that I have removed all the powder residue with Hoppe's and all the copper with Butch's, I run more patches of Naptha just to be sure I have removed all the cleaning agents (especially the copper remover, read your directions). I follow that with a dry patch or 2 and, depending on how long I figure that bore will be stored, I follow with a jagged patch of oil. If I intend to shoot right away, I use a light machine oil or Rem Oil or you choose. I like Kroil and Ballistol. If I know it will be stored for more than a week, I use the WD-40 long term storage stuff. But when I run out of that can, I am going to switch to the Hornady One-Shot due to results gathered froma couple of studies on corrosion prevention. Here is one:
https://dayattherange.com/?page_id=3667
Before I go shoot, in either case, I run a wet patch of Kroil or Ballistol and follow that with a few dry patches.
EDITED to add: I generally clean after a range session, usually about 200 rounds. If I am hunting, I will not clean but might run a patch of naptha followed by an oil patch if i've been in some fine dust or rain but usually just wait until i'm back home or the hotel before the airplane.