Interestingly enough and of note for trivia buffs, the horseman's ride, when not represented by the real thing, is the retrofitted mechanical horse that Elizabeth Taylor rode in 1944's "National Velvet."
First, take the sort of animal-in-jeopardy story made to make young audiences (and adults with tender hearts) bawl their eyes out - "Charlotte's Web," maybe, or "Old Yeller" or "The Yearling."