Either/or, doesn't matter, Ted, adn I like 320 grit. Works like a dream. Rotate your wrist (bit of cloth down the bore to catch 'grindings'),
rotate the bore 90 to 180 degrees now and then and the smoothing will automatically centre itself.
If you can find the use of a lathe that will handle your barrel through the headstock in a 3-jaw chuck, the "work" can be done in less than a minute, from chucking up the (wrap of cloth or paper around the) barrel to finishing the smoothing.
The Hawken's barrel (above post) in the bottom picture is one Taylor did on the lathe.
The bottom picture, is my work by hand from the post above, on a 20 bore's barrel, with some rifle barrels above that one.
Sometimes I do finish with finer paper or emery, 400, 600, or crocus cloth. Hemostats, medical or normal needlepoint pliars are handy for pulling the cloth plug out of the barrel. Too, dental picks are also good for that.
upload the pictures