My normal cleaning routine is to take the barrels off, and squirt cleaning fluid down each tube.
I then go and make myself a coffee to give the cleaning fluid a chance to soften the muck, then pass a bronze brush through.
This is followed by a cleaning patch on a jag. If any dirt remains, just repeat the process. For cleaning patches I usually use strips torn off a paper kitchen roll, which is soft and very absorbent.
The sort of toilet paper that’s kind to your bum does just as well.
If you use strips of rag, choose carefully.
Cotton is by far the best because it absorbs a lot of liquid, so you can tear up old shirts, handkerchiefs, bed sheets, or anything else made of that material.
Man-made fibres are less good in that many of them don’t absorb liquids very well, and nylon is particularly poor.