It's a good question.
Manufacturers sell caliber-specific cleaning kits that include brushes. This makes new shooters think they need to brush the bore, when it really isn't necessary most of the time.
I use a breech-face scraper and a jag and patch, and find that generally gets the job done. I wouldn't go too tight with the jag and patch, either... In my experience, if it takes some effort to push one into the barrel, it will take a lot of effort to get it back out.
The metal wire brushes, whether brass or stainless, are for aggressive cleaning. Black powder residue, for the most part, is water soluble. You get it wet and wipe it off, so really aggressive cleaning of a gun that has been properly maintained is seldom necessary.
The bristles on tight-fitting bore brushes have to bend back slightly as the bore brush is pushed through the bore. When you pull back on it, the bristles want to reverse, and then you have a problem. It is a non-issue with breech loaders. You push the brush all the way through the barrel and out the other end, the bristles stand up, and when you pull the brush back, the bristles reverse and lie back the other way. They can't do that with a muzzle-loader, unless you remove the breech plug, which will void the warranty on most production made muzzle loaders.
It was suggested that you give the brush a clockwise twist to facilitate reversing the bristles, and then pull it out. I have tried this, and it may help a little, but it's still a hard pull getting that brush out of there.
If you are determined to use a brush on your muzzleloader, use one with soft bristles, like nylon. If you clean your gun with hot water, the heat will further help relax the nylon bristles so they can reverse and allow you to pull the brush out. It is my understanding that bore brushes in the 19th century looked a lot like the ones we use now, but the bristles were made of natural hog bristle, which theoretically would be soft enough to reverse in the bore. However, most of the 19th century brushes I have seen were meant for the early breech loaders. The brushes were mounted on a thong, which was meant to be dropped through the bore so the brush could be pulled, rather than pushed. A thong was a lot easier for a trooper to carry than a rod.
In summary, you really don't need a brush for routine cleaning of a muzzleloader that shoots patched round balls. If you must brush, use one with nylon bristles. If you elect to use a brass or stainless brush anyway, just enter "stuck brush" in the search box in the upper right corner of your screen for advice about what to do next.
Best regards,
Notchy Bob