Here is what a couple of fairly knowledgeably people have to say...
Lilja
It is important to break-in a barrel though. The jacket material must be removed after every shot during the initial few rounds. If this isn't done the areas of the barrel that fouled will tend to pick up more fouling and it will build on itself. It is important to get a layer of powder fouling on top of the lands & grooves. This hard deposit will prevent the copper from stripping off the bullets. However, if the internal finish of the barrel is too rough the barrel will never be completely broken-in and fouling will always be a problem. Some barrels can't be broken-in.
Lilja - Break-in Procedure
For an effective break-in the barrel should be cleaned after every shot for the first 10-12 rounds or until copper fouling stops. Our procedure is to push a cotton patch that is wet with solvent through the barrel. This will remove much of the powder fouling and wet the inside of the barrel with solvent. Next, wet a bronze brush with solvent and stroke the barrel 5-10 times. Follow this by another wet patch and then one dry patch. Now soak the barrel with a strong copper removing solvent until all of the blue mess is removed from the barrel. The copper fouling will be heavy for a few rounds and then taper off quickly in just one or two shots. Once it has stopped or diminished significantly it is time to start shooting 5 shot groups, cleaning after each one. After 25-30 rounds clean at a normal interval of 10-25 rounds. Your barrel is now broken-in.
For best results, of course, it is necessary to 'season' it and use proper cleaning equipment. We like the Dewey coated rods, a good bore guide, copper/bronze brushes and cotton flannel patches, the appropriate size to keep that jag and rod in the middle of the bore. You will need a good bore solvent, like KG 3, Shooters Choice or CR-10 to loosen the fouling, followed by a scrubbing with Holland's Witches Brew or KG 2. After cleaning, nullify the solvents with rubbing alcohol and patch dry. Finish with Tetra Gun Oil, KG 4 or Holland's Bbl Break-in Fluid. Never shoot a dry bore as this will greatly promote copper fouling.
Krieger - Break-in Procedure
With any premium barrel that has been finish lapped -- such as your Krieger Barrel --, the lay or direction of the finish is in the direction of the bullet travel, so fouling is minimal. 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 released into the gas which at this temperature and pressure is actually a plasma. The copper dust is vaporized in this gas 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.
Barrels will vary slightly in how many rounds they take to break in because of things like slightly different machinability of the steel, or steel chemistry, or the condition of the chambering reamer, etc. . . 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 the same 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. (Chrome moly and stainless steel are different materials with some things in common and others different.) 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 clearing procedure are really the same except for the frequency. Remember the goal is to get or keep the barrel clean while polishing out the throat.
Finally, the best way to break-in the barrel is to observe when the barrel is broken in; 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.