***Some petroleum based lubes are compatible.***
After all, mineral oil, paraffin wax, petroleum jelly are petroleum based and won't cause any problems.
Petroleum is a natural product but a lot of different derivatives can create tar.
The main ingredient in Ballistol & Bore Butter & Wonder lube is mineral oil, and there's wax in BB & WW too.
Without specifying which lube, then people will ask "Oh I thought that all petroleum based lubes are no good and will cause problems?"
Only some petroleum products cause tar, while others can lubricate quite well.
Read the labels of common patch lube products and you'll find that many contain some amount of mineral oil.
They can make a mess but putting some of them over the ball can also be effective in keeping fouling soft inside the bore.
I think that it depends on the ingredients of the lube that's being used.
People have reported that using a very thin bead of lube on top of the ball works for them such as Vaseline, and the original Gojo waterless hand cleaner without pumice.
Some people pack the top of the chamber with a harder lube and that works very well too.
Personally I don't use lube with American Pioneer Powder because it's not recommended.
When I did try BB lube over the ball it certainly coated the inside of the bore making it quite greasy.
However I prefer to use APP without any lube and just swab the bore after 40 or more shots if needed, loading 30 grains of powder.