That's a little tricky, not knowing what the saddle has been treated with. But if it's oil that's lifting oil-based dye, I'd try to remove as much as poss by applying a SMALL amount of known oil (maybe PURE neetsfoot oil, NOT the compound) on a clean towel and rub until it comes up clean on the towelling. Then I'd seal it with a CLEAR pastewax or shoe polish, and keep the wax renewed as needed to maintain the seal. None of this will do any harm to the leather, and will probably help it. Stick to a good wax without harsh solvents, I use and recommend Liberon Black Bison CLEAR wax (you don't want to add more dye) on fine furniture, and have used it once in a while on leather with no ill effects, it has very mild solvents and actually smells good, unlike Briwax, Treewax, or SC Johnson Pastewax NONE of which I'd use on leather if I have a choice.
Also, wear black shorts...