For the breastplate, throw it in the washer on delicate cycle with Leather Therapy or PePeDe detergent.
For the rubber reins, if they're just dirty, plain water or water with a touch of vinegar should remove the grit. If they're gummy and "melty", they're done for and you'll either have to throw them out or pay for rubber replacement.
Toothpaste! Yes, use toothpaste and a stiffbrush to scrub in the paste. Let sit for a litle bit, then rinse with a wet sponge. Don't be too liberal with the paste, it'll make for a lot of work getting it out.
Any type does a good job. Since you do not want the elastic to whiten, I suggest you try a small spot on the undeside first.
I wash all my daughter's horses leather equitation boots, fleece lined girth and elastic breast plate in the washing machine with Leather Therapy. I put the boots and the breast plate in a lingerie bag. Everything comes out great.