Some time back, I remember a comment from a barrel manufacturer, something to the effect that he did not recommend a patch over a brush with JB, because it could work like a lap, that the pressure might actually promote more action on the barrel itself than is desirable. One of the reasons that I shy away from abrasives, and use them cautiously is that in producing that bright, shiny bore, one may over smooth it, creating a jacket fouling problem that requires redoing the break in. If I were going to do the job these days, I would lightly oil the bore, and the patch, rub JB into the patch, use my regular jag, take some care at the throat, shot stroke up the bore, and pay particular attention to not spending any more time than necessary at the crown. One thing that I came across, by accident, a while back, was that light gun oil does a better job of floating lapping compound out of the surface of aluminum rings than things like alcohol or water and detergent based cleaning liquids. I think that this is the reason for the Kroil and JB routine. The Kroil suspends what the JB has cut from the bore surface. I suppose that I should add that in the past I have used abrasives in all sorts of ways that I would not repeat today.
A long time ago, benchrest shooter and gunsmith Mickey Coleman wrote that wrapping a little 0000 steel wool around a worn out brush, and using it with plenty of solvent would not hurt a bore and would get it really clean. I have done this, and he is correct. In fact, I would suggest that if properly done, it may be one of the better ways to deal with a badly neglected bore. As with other aggressive methods, be careful at the throat and crown.