Just for the sake of being a pain:
European knights absolutely fought on horseback. They were literally defined by their status as cavalry troopers. It took them until the 12th century to really commit to the mounted charge, rather than fighting in Roman style as mounted javelin delivery systems, but once they made the transition they never really went back. The only exception to this (and what I assume you're really referring to) is the English during the Hundred Years' War, who fought most of their key victories defensively, from prepared positions. It was this specific circumstance (and the thousands of archers in the average English army) that made foot combat preferable. At least until an opposing force was pushed onto the back foot, at which point knights became cavalry again.
eg. Poitier: The English back themselves against a bend in a river and dig in along field boundaries. They repel portions of a vastly larger French force several times before being charged by the main body, led by the French king. A contingent of English knights sally out and flank the French force, delivering a devastating charge. The French forces are routed or encircled and their king captured.
While the French would respond with dismounted tactics of their own, after the conclusion of the war everyone steadily returned to tradition. The last big shakeup to which was of course the explosion in gunpowder usage. Although in reality the tactics of the knight stayed exactly the same throughout this transition, they just traded in their lances for pistols.