Much of it depends on the kind of barrel, a match grade stainless should be broken in and every barrel maker has their own procedure. Just shoot a chrome or nitro barrel, then clean the copper fouling out when you get home.
match barrel break in-
Barrel maker websites are a good place for info , Krieger, Lilja, Hart, etc
Lilja has many good articles on his site.
I foam my AR barrels with foaming copper cleaner, let it sit for 10 minutes and patch it out, if it comes out blue, foam it again and again until the patch comes out white.
Clean the solvent out with alcohol then run a patch with light oil wd40 or rem oil. Then a dry patch to remove all excess oil, it can cause a pressure issue if left in the bore. After the barrel stops copper fouling I clean less and less, I have a few stainless match barrels with 3-400 rounds on them, some ARs may have over a thousand. I can check for copper fouling with a borescope, it makes it easy for me but let the barrel tell you when it needs to be cleaned. As long as it's accurate leave it, when accuracy drops off it's a good sign there is too much copper or carbon fouling.
I run mobil 1 on the carrier rails, bolt and trigger pins and sear but do not let any enter the bore. I oil before I start shooting, every trip to the range.
I'll look around for other posts.