Pack animals are still in use today; the wagon and cart never superseded them. The wheel did spread, but quite slowly. Motorised transport had more impact on pack animals than animal-drawn wheeled transport, and of course also had a large impact on animal-drawn carts and wagons. For past discussion on this, see my answer in:

• https://www.reddit.com/r/AskHistorians/comments/j2en2t/when_did_people_stop_using_pack_animals_and_start/

For academic sources on competition between pack animals and wheeled vehicles, see

• Bulliet, Richard W., The Camel and the Wheel, Harvard University Press, 1975.

• Bulliet, Richard, The Wheel: Inventions and Reinventions, Columbia University Press, 2016.

which were cited by myself and u/Chamboz and u/Aithiopika in:

• https://www.reddit.com/r/AskHistorians/comments/e2locu/why_was_wheeled_transport_replaced_by_animals_in/

Bulliet (2016) also discusses competition between pack animals, carts, wagons, palanquins/litters, and human vs animal power.