CLEAN
Water ... room temp, NEVER warmer to hot!
SOLVENT
If I need to use a solvent, Butch's 'Bore Shine' for black powdah is phenomenal, best I have ever used!
STORAGE
Short-term: Ballistol.
Long-term: Birchwood Casey's 'Barricade' product.
Ultra-Long-term: RIG grease.
^^^^^
This is probably the best advice if using Blackhorn.
When you first start to clean the fouling push a dry patch all the way through the bore a couple times. That will remove most of the loose fouling. Then start using a solvent.