Oh, they did. Agincourt in 1415 is a good example of the shortcomings of the heavy mounted knight, though. Henry V, with a force of about 6,000 mostly lightly-armed archers, met a French force of about 25,000, mostly armored cavalry and some infantry. The combination of the English longbowmen and the extremely muddy conditions of the battle (and the fact that the French couldn't agree on who was supposed to be leading them) meant that, when all was said and done, Henry had taken about 200 casualties, and the French, nearly 6,000. Once the knights were unhorsed, they were in serious danger - the mud made it difficult to get back up, especially with others on top of you. So they were particularly vulnerable to the English footmen rushing in to slip a dagger between the joints of their armor. And probably for as many as were killed by the English, there were just as many who literally drowned in the mud.
That lack of mobility you cite was also apparent during the Crusades - the light cavalry of the Saracens gave the Euros fits. The Mongols, et cetera, did indeed use a different style of warfare, but that was the European model. With the introduction of the longbow, and later the cannon, the days of the heavy mounted knight were definitely numbered - by the late 15'th century, they were pretty much gone, abandoned in favor of archers, light infantry, and some much lighter cavalry.