From a well known barrel maker
With any premium barrel that has been finish lapped, the lay or direction of the finish is in the direction of the bullet travel, so fouling is minimal compared to a barrel with internal tooling marks. This is true of any properly finish-lapped barrel regardless of how it is rifled. If it is not finish-lapped, there will be reamer marks left in the bore that are directly across the direction of the bullet travel. This occurs even in a button-rifled barrel as the button cannot completely iron out these reamer marks.
Because the lay of the finish is in the direction of the bullet travel, very little is done to the bore during break-in, but the throat is another story. When your barrel is chambered, by necessity there are reamer marks left in the throat that are across the lands, i.e. across the direction of the bullet travel. In a new barrel they are very distinct; much like the teeth on a very fine file. When the bullet is forced into the throat, copper dust is removed from the jacket material and released into the gas which at this temperature and pressure is actually a plasma. The copper dust is vaporized in this plasma and is carried down the barrel. As the gas expands and cools, the copper comes out of suspension and is deposited in the bore. This makes it appear as if the source of the fouling is the bore when it is actually for the most part the new throat. If this copper is allowed to stay in the bore, and subsequent bullets and deposits are fired over it, copper which adheres well to itself, will build up quickly and may be difficult to remove later. So when we break in a barrel, our goal is to get the throat "polished" without allowing copper to build up in the bore. This is the reasoning for the "fire-one-shot-and-clean" procedure.
Every barrel will vary slightly in how many rounds they take to break in For example a chrome moly barrel may take longer to break in than stainless steel because it is more abrasion resistant even though it is a similar hardness. Also chrome moly has a little more of an affinity for copper than stainless steel so it will usually show a little more "color" if you are using a chemical cleaner. Rim Fire barrels can take an extremely long time to break in, sometimes requiring several hundred rounds or more. But cleaning can be lengthened to every 25-50 rounds. The break-in procedure and the cleaning procedure are really the same except for the frequency. Remember the goal is to get or keep the barrel clean while breaking in the throat with bullets being fired over it.
Finally, the best way to tell if the barrel is broken in is to observe the patches; i.e. when the fouling is reduced. This is better than some set number of cycles of "shoot and clean" as many owners report practically no fouling after the first few shots, and more break-in would be pointless. Conversely, if more is required, a set number would not address that either. Besides, cleaning is not a completely benign procedure so it should be done carefully and no more than necessary.
From another well known barrel maker
The age old question, "Breaking in the New Barrel". Opinions very a lot here, and this is a very subjective topic. For the most part, the only thing you are breaking in, is the throat area of the barrel. The nicer the finish that the Finish Reamer or Throating Reamer leaves, the faster the throat will break in.
Shoot one round and clean for the first two rounds individually. Look to see what the barrel is telling you. If I'm getting little to no copper out of it, I sit down and shoot the gun. Say 4 - 5 round groups and then clean. If the barrel cleans easily and shoots well, we consider it done.
If the barrel shows some copper or is taking a little longer to clean after the first two, shoot a group of 3 rounds and clean. Then a group of 5 and clean.
After you shoot the 3rd group and 5th group, watch how long it takes to clean. Also notice your group sizes. If the group sizes are good and the cleaning is getting easier or is staying the same, then shoot 4 - 5 round groups.
If fouling appears to be heavy and taking a while to clean, notice your group sizes. If group sizes are good and not going sour, you don't have a fouling problem. Some barrels will clean easier than others. Some barrels may take a little longer to break in. Remember the throat. Fouling can start all the way from here. We have noticed sometimes that even up to approximately 100 rounds, a barrel can show signs of a lot of copper, but it still shoots really well and then for no apparent reason, you will notice little to no copper and it will clean really easy.
This is meant as guide lines only. There is no hard and fast rule for breaking in a barrel.