I've used JB bore cleaner paste for years - everything from bore cleaning to imparting a dull sheen finish to polished stocks.
For bore cleaning, you take a cleaning patch tightly wound around a bore brush and saturated with the past and bore cleaner, such as Hoppes. The bore should be wetted with Hoppes (or equilivent). PULL the brush from chamber to muzzle.
Don't do it the other way, or you can mess up the muzzle crown. Do this several times and see how black the patch is! Renew a wetted and JB'd patch, swab bore and re-wet, then do it again. Repeat this a half dozen time and that bore will be shiny clean.
Shooting soft lead .22 bullets impregnated with JB would probably lap the bore to a degree, but the NECO lapping paste would be more effective.