I've used aniline dyes on bows many times... water, alcohol, and oil soluble, and all sorts of colors... reds, browns, blue, black, yellow, orange, etc, and I often mix at least two of them to get the look I want.
I won't use water based dyes any more because it raises the grain of the wood. It can be dealt with somewhat, it just seems unnecessary to me. You can reduce this effect by first repeatedly wetting the wood with plain water and sanding it, but I'd rather just use alcohol or oil based dyes.
Oil based anilines may be harder to dissolve in, or may not dissolve properly in some 'oils'. Which oil or petroleum based solvent that's best used with it depends on the look you're trying to achieve, the finishing process used, and it takes time and testing to get it right. Will the dye be IN the finish, or in a solvent, with the finish applied over the dyed wood? Lots of options.
Multiple base colors can be combined to get the exact color you want, and you can also fade/blend from one color into the next right on the wood. I've found them all to be colorfast enough, and can only recall one bow that faded noticeably, though it still looks good. I made it so long ago that I can't remember which type of dye I used on it.