When i return home from shooting a rifle, the first thing i do is remove the breech plug. Then i run a dry patch down and back, and toss it.
Then another dry patch down and back; turn it inside out, and run it down and back, and toss it.
Then i soak a patch with solvent, and run it back and forth in the barrel, a few times.
Then a clean dry patch is run down and back, which soaks up the solvent.
Then a clean dry patch is run a few times down the barrel; it is turned inside out, and run a few times down the barrel. This patch usually is quite clean on the reverse side when finished.