I've never been bit by a dog, and generally don't tolerate them getting close. Nor have I been bit by a wild animal.
So, the first would be prevention.
I probably should carry sterile gauze, but don't. I believe that I could use my own clothing for dressings in many situations. Stop the immediate danger, then deal with infection later. And, if there is a serious situation, most drivers will stop, as well as aid from neighbors, or others you might encounter.
I saw the mention of "Combat Gauze" recently. That sounds like pretty wild stuff!!!
I'm not sure how far one could ride. If you rode a few miles, you could probably ride all the way home. But, it does depend on the extent of the injury. I know there is a "cat scratch disease", but fewer canine specific diseases that affect humans, I think. Nonetheless, deep penetrating wounds can bring bacteria into places you don't want it.