I'm hooked on foaming bore cleaner. I think a few different brands are available; I'm using the Breakfree brand. Squirt it in the bore, let 'er sit for a few minutes, and then swab with a couple patches. It's the consistency of shaving cream, and it thoroughly coats the bore when applied. Seems to attack copper pretty aggressively, based on the amount of blue that's present on the patch after swabbing. Most of my cleaning with the foam is on rifles, and those seem to produce more copper fouling than the 250.
The foam makes the cleaning regimen easy. I squirt the foam, swab it out, follow with an oiled patch, and finish using a dry patch.
The upshot is I don't use a brush. However, when I do use a brush on other bores, it's either a bronze/brass or plastic bristle, both Hoppe's brand.