The English longbow was estimated to have a draw weight of between 45kg and 82kg (100lbs to 180lbs)[1] but of course was much bigger and couldn't really be used on horseback.
[1] https://interestingengineering.com/lists/the-most-powerful-b...
[2] https://centerofthewest.org/2017/07/21/plains-indian-weapons...
In the Homeric poems, Iliad and Odyssey, composite bows are described, e.g. used by the very skilled archer Odysseus.
Moreover, even older descriptions (from the second half of the second millennium BC) of composite bows can be found in the myths preserved on the tablets from Ugarit.
It is true that an Asiatic composite bow over a certain draw weight (depending on design) can impart greater energy on the arrow than a self-bow (long bow) of the same draw weight. The force-draw curves are different, and some of these bows have very long draws in comparison.
The site the interestingengineering references, itself does not contain references. There are odd discrepancies, such as the section on stringing the bow. Past a certain draw weight, you can no longer string the bow using the one leg or sitting method. The Manchus used equipment off the horse to string their bow, and kept them in cases while strung, to protect them.
Europeans did not bring horses to America until after the Mongol empire had already collapsed, so the people there (whether called Comanches, Shoshone, etc.) couldn't even have any horseback archers. :P
"would have fared Very Badly (or perhaps Not Applicable.)"
Without that, I think the Mongols would easily win. They had a huge army that rapidly conquered the largest contiguous land empire ever. They had powerful bows, better armour, very effective tactics, but they were also excellent at sieges. Every country they conquered, they assimilated the best ideas and technology they had, which made them able to shift to very different types of warfare. They were capable of strategic maneuvering on a scale that wouldn't be seen again until WW2.
Their weakness was primarily in their rules for succession. If the death of a leader requires you to abandon all campaigns and moving all generals to the capital, after which the empire still fragments into several pieces, that's a bit a of a problem.