Running shoes (Brooks) because I get off and run/walk with my horse up and/or down mountains. One of my endurance friends used to swear by her Arriat boots, but she never got off her horse except at the checks. Don’t think I’d want to be hand jogging my horse down a mountain unless I was in my running shoes.
Also wear half chaps over my Irideon or Karrots tights (that have a homemade pocket sewn in to carry my phone on me).
My stirrups are the EZ Ride endurance type with the cage. https://www.easycareinc.com/Other_Products/E-Z_Ride_Stirrups.aspx Absolutely love them. On bad weather rainy rides, I can duct tape plastic on the front of the cage to keep my feet dry (and warm). Mine are an older pair (maybe about 8-9 years, and thus didn’t have the type of pad that comes with them now. I cut out a piece of impact padding and used that for the stirrup pad - easier on the knees and foot, and not as “grabby” as the one with a rough surface. I still have a pair of aluminum endurance stirrups that are open in the front, but I was a little leary of using them, especially since I wear running shoes. They are on a secondary saddle now that I use rarely.