Alternatively it could be argued that increasing temperatures and abundance of food favored smaller animals (including our very low-tech ancestors), which pushed the proverbial mammoths out of their ecological niche and into the oblivion.

The smaller the animal, the larger skin_surface:body_mass ratio, thus higher heat loss. The smaller the animal, the shorter the intervals between feedings it can tolerate. A large-bodied animal can go for longer period of time without food, especially if equipped with slow metabolism. In contrast, smaller body and faster metabolism requires frequent feeding, but also allows an animal to out-compete slower, larger ones both in food search, and by out-breeding them.

For any given body plan, an animal can only cover distance roughly relative to its body size; the smaller animal, the harder time it will have to find sufficient food on sparse, hardy cold-climate vegetation. On the other hand, once denser, warm climate vegetation takes hold, it allows smaller herbivores to live in higher density, and with higher breeding rate.

If I remember my biology right, the more an animal has specializations for particular environment or niche, the slower it will adapt to changed circumstances, both in body build, and in behavior & seasonal migration patterns.

Last but not least, a predator population requires certain density of prey to be sustainable. Once plant food becomes abundant due to warming up, and herbivore population grows, the population of predators can be established, and cull the large bodied animals unadapted to heavy predation.