On my last engine rebuild (1985 Jaguar 4.2), my machine shop advise the following: 30-weight NON-detergent (hard to find anymore!) for 500 miles, change oil & filter; 30-weight NON-detergent for another 1,000 miles, change again, then change to the permanent oil (I'm using Redline synthetic).
The machinist explained that the rings need to wear in to the piston bores and the bearings to the crankshaft surfaces, and the slightly higher amount of friction from the 30-weight non-detergent will allow this to happen.
Also, I didn't exceed 2,500 rpm for the first 500, and 3,000 for the next thousand. Then, up to 5,000, I only used short bursts of full throttle.
At 13,000 miles since the rebuild the engine does not burn a drop of oil, runs incredibly smoothly, and is almost undetectable at idle.
Where the factory can do the run-in on a test bench, you have to do it yourself, and be gentle, but not too gentle. No long freeway runs. Vary the speed constantly.
If it's a clutch, don't coast to a stop, but slow down in gear, allowing the engine to overrun a little. The whole idea of breaking in is to use the engine over a broad range of the rpm band, but to stay well away from the redline at first.
I believe that in one of his books, Geoff Healey wrote that when his father was building the Nash-Healey, they had problems with the Nash six-cylinder engines they were buying. They imported them in crates, dry, installed them in the cars, and filled them with quality Castrol oil. They found problems with excessive oil consumption for several thousand miles due to piston rings not seating, until they learned that Nash used a lighter weight break-in oil in the cars sold in the States, which the dealers would fill the crankcase with during the predelivery service. Once Healey started doing this the problem went away.
I also recall reading about the early Jaguar 3.8 sixes in the E-types, which were notorious oil-burners, that some mechanics would shoot some grinding powder into the intakes to scuff the rings and bores. Not that I'd want to have that done to MY car, you understand . . .