Animating blood dripping down surface of mesh

Can anybody please point me in some kind of direction for animating blood/tears/whatever dripping down the surface of a mesh? Maybe particularly the “correct” method a studio would actually use?

I’ve tried a few approaches now that all fail for one reason or another. Dynamic paint in vertex mode doesn’t look good/can’t figure out how to do it over an existing texture/requires adding >9000 subdivisions to still look bad.

Dynamic paint as image sequence seems like it would be the would be the way to go when I use it on a plane but when I try to do it on a characters face it just makes a tiny little splotch that doesn’t move with the brush at all.

Someone suggested texture paint but I have this weird problem when I try to paint on an existing texture where the color shows up black and white and doesn’t appear in rendered view- a problem that can only be mitigated by painting on a solid color and plugging the image texture in afterwards. More importantly than that I don’t see how I can record my brush strokes to make the thing actually animate.

This seems like such a basic thing that it’s impossible to find information on?