Thought I had something in the Tilley Files

Page 76 of The Parker Hale Gun Catalogue No 67 details Ball Burnishing as:
'The Ball Burnishing process consists of forcing very finely graded hard steel balls through barrels that have been selected from target quality production for all aspects of their internal condition.

The ball is first selected to match the initial bore size or gauge of the individual barrel; it is continuously passed through the bore and in succession balls of slightly greater size are used to level out any internal imperfections, thereby imparting to the lands a homogeneous work hardened surface which assists accuracy through improved parallelism whilst effectively increasing the resistance of the lands

Ball Burnishing enlarges the average bore by approximately one half thousandth of an inch, an almost infinitesimal amount which in no circumstances can cause detrimental effect.

New barrels are officially acceptable today when made between .301" and .304" gauge, but 98% of new barrel production is found to gauge between .301" and .3025".

After Ball Burnishing, these sizes may be increased to a maximum .30275" gauge.

Barrels when sectioned show that Ball Burnished samples have a dark mirror-like surface of the lands, whereas others not burnished although well-rifled disclose innumerable cross-cuts.
Even the finest boring leaves a series of tiny cross-cuts which tend to tear the metal from the bullet envelope during its passage up the barrel.
Not only does it improve the surface as stated above, but it is to be recommended mainly on new barrels to improve their initial levelness, to extend their normal life and to minimise the collection of metallic fouling.

It is not recommended for barrels that have been so worn as to round off the corners of the lands, for this reduces the area of contact to the ball in the burnishing process.

Nevertheless, a barrel that is not too badly worn can often have its accuracy improved by Ball Burnishing.