There’s a product out there called "Casting(s?) by either Clairol or L’Oreal. It’s temporary (meaning it washes out in 10 or so shampoos). If you’re just starting out, try that. Don’t try the permanent stuff (and anything with peroxide in it is usually permanent) until you’re used to the process and like the results. Don’t go too dark! It will “flatten” all your, er, your horse’s, highlights and the hair will look like one boring blob. Another advantage of the temporary dye is that it gradually washes out so you don’t get roots.
Also, be forewarned that dying a dark shade (something other than blond) may color your scalp, er, your horse’s mane and tail base. That’s okay. Even with the premanent stuff it washes away in one or two shampoos.
When you do decide to use permanent stuff, be aware that it will lighten over a week or so as it gets exposed to air. The color you get after you first dry your hair will look very severe, but it will calm down.
Seriously for horses… a couple friends of mine have had good luck using color enhancing shampoos (the kind sold for people) to cover up minor sun bleaching of black/dark brown manes and tails. There’s no committment since it washes out, and you don’t have to be nearly as careful as with the more potent stuff.
A note of warning: even the temporary color products or the color enhancing shampoos can end up permanent if your hair has already been chemically treated. So they’re not a solution for overly highlighted hair! Believe me, I tried, and the results were not pretty!