ThisI think this is pretty spot-on for The Family Tree by Sheri S. Tepper from 1997.
All the points match. The "riding beasts" thing isn't a big plot point; humans are used as beasts of burden and as tools most of the time.
The book is split into two timelines - the one you're describing is the future. There's also a contemporary timeline explaining how we genetically engineered animals to have greater intelligence, basically an "uplift" novella.