To clean a dirty bore after shooting, I do the following:
- Spray carbon remover down the bore - I use Mercury/Quicksilver power tune (carburettor cleaner). This may not be necessary and previously I used a oily patch (kroil or good penetrating oil).
- Dry patch.
- Patch with Montana X-Treme copper cream. (used to use gold medallion, but copper cream is better and less abrasive)
- Dry patch.
- Repeat copper cream patch followed by dry patch as necessary - rarely needed for hunting rifles, more for competition rifles due to number of shots fired.
- Oily patch.
- Exterior wipe down.
- End.

Before shooting, I remove oil from the bore:
- dry patch
- patch soaked with Iso-propanol (IPA - pure rubbing alcohol - used for cleaning circuit boards etc and leaves no residue) (or lighter fluid). Can be sprayed down bore from aerosol
- dry patch (two if IPA spray used)
- sighter/ fouling shot. Some of my rifles shoot about 0.5cm high after this process on the first shot.

NEVER had a problem with carbon, copper build up, or anything. Never had to use a brush either.
I used to use a foam similar to wipeout until I ruined an expensive rotary magazine box, housing and synthetic stock (older injection moulded type - crap but nice shape).

Just my 2cents worth...