Everyone's got their own theories about break-in or even
not to break-in. Varmint Al just runs 50 strokes with Flitz before every firing for the first time, and will repeat with JB if the bore still seems to foul. No shooting required.
I don't consider break-in a critical issue in most instances. I do know that
some barrels will burnish and foul less if you clean them completely between shots for the first 10 to 20 rounds, though most of the improvement seems to happen in the first 10 if it's going to happen at all. Others, and I've had a couple, start as bad foulers and stay that way right through any break-in ever devised, and staty that way until you do something more extreme, like firelapping. The jury is still out with me on getting the barrel to take a set with the first few rounds (Howa's distributor's theory).
The idea behind cleaning between every round is that fouling from the previous round collects on and covers a surface that is grabby and that needs to be re-exposed for burnishing by the next shot. I've never figured out the advice to fire multiple rounds between break-in cleaning or to graduate from cleaning every round to every two or three then every five, etc. It seems to me all that does is guarantee the roughest spots are well protected from burnishing as you go on to subsequent rounds before cleaning. Maybe someone else sees something going on there that I've missed, though.