Are we talking about neglected pipes? Maybe for years? (Don't judge me.) A very small flat-bladed screwdriver can be used to start scraping on the mortise gunk buildup. You're digging into a corner there (flat-bottomed hole, right angled walls). In my experience, you can use swabs and pipe cleaners till doomsday and they'll still come out brown tinged. A screwdriver deftly applied speeds up the process.
But I'm slowly coming over to the warm water douche method of freshening a pipe. Works instantly, it seems. Have yet to try it on $200+ pipes, but why not? (Cowardice comes to mind, but I'm a big boy now, Mommy.)