I made my own lapping kit.
Water based diamond pastes.
https://www.ebay.co.uk/itm/7x-THK-Diamond-PLUS-5g-WATER-BASE-polishing-lapping-paste-pastes-compound-0-25-5/153995265671?hash=item23dad54687:g
MAAAOSw32lYwArD
And then I use a microfibre swab on an unjamming rod, and spin it through the barrel with a drill.
You'll want 2 swabs per grit level.
Use one to apply and polish, then the second with water to clean the barrel.
You will want an old barrel to practice with first, and used swabs can be used multiple times so keep every grit and it's 2 swabs separate from each other.
Grits are worked out in microns, bigger numbers are bigger abrasives. So a 5 micron is more coarse then a 1 micron.
Pick a middle grit to start with 2-3 do a few passes of the barrel with the drill, then clean it up. If there are big scores that haven't been removed think about moving to a higher grit, or doing multiple 3 micron grit passes. Once it looks good at that stage then clean it all again and start to go lower. 2 - 1- 0.5 - 0.25. There will be some haze to the finish that should be clearing up with each lowering of the grit, and by the time you get to 0.5 micron it should be looking polished - Free or scores, and be giving a straight reflection. If the reflection isn't straight then you need to inspect the barrel and make sure it's not warped and that you haven't done damage at a higher grit by keeping it spinning on one spot too long.
When complete the swabs will be black and horrible, but they will still work on future barrels.
The finish will only be as good as you. If you are not consistent with the drill and pull speed you will have issues, Likewise the swabs need to be made right so they give a nice fit that has good contact but isn't tight.
