Looking at barrels as you get them, they are either lapped by the maker, as custom barrel makers do with their barrels, or they are unlapped, like you get when you buy a factory rifle.
Custom barrel makers like Hart, Lilja, Shilen, Kreiger, and others, all lap their barrels before you get them. For these barrels, there is no break in required as they are as good as they will ever get.
Factory barrels that aren't lapped have tool marks inside that should be polished/smoothed (lapped). I use JB Bore Paste to poilish (lap) my factory barrels, 100 strokes with a tight fitting patch loaded with JB. After doing the polishing, these barrels are as good as they will get, and no break in is required.
Don't confuse rifle break in with barrel break in. A new rifle will need a few rounds through it for the barrel and action to settle in the stock so it shoots better. But, the barrel does not need any breaking in if you do what I decribe above.
I use JB Bore Paste as it is slightly more abrasive that Flitz, and does a better job of polishing out the tool marks in the bore. Flitz is a polish with very little abrasive in it, so it does polish, but doesn't remove the tool marks like the JB does. I've used both of these products for 30 years in my knife making, so I have experience with them. The Flitz would be best used after using the JB for that final bore shining.
JB was compounded for rifle bore lapping and is sold by Brownell's, a gun parts supplier.
Martyn