I was going at my saddle with some toothbrushes the other day and wondered if I was the only one that did this
my friend thought I has clearly lost it, but it works! And that saddle has so many crevices!
My most used materials on tack would be (aside from toothbrushes) a soft cloth, small sponge, and warm water. Where I live it is often damp and leather doesn’t dry out so much when stored in the stable. So conditioning is not needed often.
I would go at it with warm water, if that isn’t doing the trick, then I’d reach for some castile or dish washing soap.
I’d then let the saddle dry and see how it feels. If it needs a conditioning, then I’d do that. When I use leather conditioner I often massage it into the leather with my hands. I let it sit a bit, then wipe/buff the leather to remove any excess.
Then just wipe it off with a cloth or with water as needed. You can also get stirrup covers to prevent dirt from the stirrups from getting on your saddle. Although mine just gets dirty from being within close radius of my horse it seems!