Pecard makes a number of products. It is their "Antique Leather Dressing", that you want.

My wife swears by Saddle Soap, but, she uses it slightly damp with a spounge on Horse Tack and riding boots.
(Dirt, salt, and horse sweat are different than what you are encountering).

I have found Hoppe's #9 on a cleaning patch to be very effective in cleaning verdigris from brass, while leaving a nice patina.
Just limit its contact with the leather.

A weak solution of water and Dawn detergent on a wash cloth might lighten stains on cloth loops. Brushing with a clean toothbrush might take debris off. Don't get things too wet.