I used Vetrap to secure my splinted dislocated pinky so that I could ride at Pony Club championships this past summer (Warmblood gelding decided he did NOT want to go to the jog up with everyone else and would prefer a private audience with the vet after my trip to the ER thank you very much). Vetrap plus cutting off the pinky of one of my Roeckl gloves and I was “good to go”.
I keep both a human emergency bag and a horse emergency bag in my tack room. When I haul horses both bags get moved to the trailer. I chose bags because of easier foot transport out into pasture or onto a corse or arena. and everything can go with me at once hung over a shoulder without wasting time gathering items.
I also have a lot bigger selection of items in the tack room. The lists are pretty lengthy, and there are good posts above, but the picture is my answer to the “what do you keep it in” question.
Smaller bag is human; larger bag is horse. One thing I do have in both is Blood Stop trauma gauze and packs. They really do the job and there are many available brands these days. Expensive, but so is your horse or child.