Factory barrel or a premium aftermarket barrel?
Factory barrels are pretty poor barrels compared to premium barrels. They are drilled, reamed, and button rifled, or they are hammer forged. Either way, they get no attention to interior finish, so they tend to be kind of rough. These are the barrels that may benefit the most from break-in, but deserve it the least.
Premium barrels are typically lapped to very fine microfinish and only need minimal break-in to burnish the interior. Your first sighting in session will break them in just fine.
I had never heard it called break-in, but 30 years ago when I bought broach-cut barrels, I was told to shoot and swab until the barrel quit shaving copper from the bullets, about 25-50 rounds. Nowadays that is called break-in, I don't know if we had a name for it back then.