One usually walks through an area which is enclosed somehow, e.g. with walls, railings or trees. (Also one enters the area at one point and exits it at the other.) Would the leaves constitute such an enclosure?
One would be walking through the leaves if there were enough of them, but walking through a path sounds very odd to me. I am surprised the dictionary gives it as a possibility. Walk the path without any preposition is possible but unusual. I can't think of an example but I'm not going to bother because I don't think it is correct in this case.
I'd say "along the path" if a casual stroll were involved. Or perhaps up or down. One of the latter might make sense if there were a goal or direction stated or implied: We walked down the path to the lake. He walked up the path to the door of the house. Don't go down that path; it leads nowhere.