There is no reason to not think that people from Europe could have made it to the Americas in small sea crafts but not bringing animals - the journey was far too long to have space for provisions for them
Heyerdahl did his sea voyages in crafts made to primitive designs to show that people could have made the journeys all around the globe taking with them their own histories.
The cave paintings in South America could well have been made by a people who travelled there from countries where horses were well known but not essentially that they took them with them or still had access to them
The gaited horses of the Mongolians spread to all parts of Europe. There are historical references to horses that moved at a gait or amble from as far back as 6000BCE - attached link to info on them in European History as the references to gaited horses in Europe pre-date those of the Chinese