If you grab an animal with a lead, and go up far enough to pull them off the ground, they will usually take fall damage when they touch the ground, regardless of how slowly they fell. This is likely due to the fact that fall damage is calculated by time in air, or time falling, and not by speed of descent like it should be (in my opinion). This is also related to the bug where players take massive fall damage in boats if the boat happens to land somehow (such as sailing onto a dock made of slabs).
It would really help if this bug was fixed, because I lost two horses trying to pull them out of a 7-block deep hole, and having them die as soon as they touch the ground, even with the lead still attached.