From the beginning of Spanish exploration of Mexico and North America they'd been losing horses left and right ~ by this time the Indians at Cahokia were already riding off toward the great herds of buffalo on the West bank of the Mississippi ~ and becoming the Sioux and Cheyenne we think of today.
North America was made for the horse. They prospered beyond all belief.
I don’t remember reading of any horses getting loose till after the Puebelo revolt in 1680, Santa Fe. Hogs yes! Horses, no.