There are sort of two answers to the titular question that can be found by looking at one of the last major conquerors to emerge from the steppe and use horse archers: the Qing dynasty in China, where the answer (depending on your perspective) is either that cannons and walled cities necessitated a move away from standard horse archer tactics in the mid-17th century, or that long periods of peace caused the tactics to fade into obscurity and be entirely obsolete by the time they might be needed again. The Qing dynasty was built by the Manchus (a Jurchen steppe tribe coalition turned ethnicity) working in conjunction with other steppe tribes, including several Mongol tribes. I should mention that their tactics were extremely similar to the Mongols', based on hunting methods that involved horsemen encircling a target and whittling it down with recurve bows. The Manchus made a series of conquests taking over Ming China in the early to mid 17th century on the strength of the superior mobility of their nearly all-cavalry army, the core of which was horse archers.

Initially, the Manchus devastated the Ming dynasty's forces because of the superiority of the mounted troops - despite the fact that the Ming soldiers were armed with gunpowder weapons. It didn't make a significant enough difference faced with the Manchus' mobility and adaptability. An example of the way the Manchus used this advantage is the Battle of Sarhu, where the Ming forces split into three armies and attempted to outflank and surprise the Manchu forces. The Manchus responded by moving their whole army to engage each of the three armies one at a time. Although they were the smaller force, the fact that they could cut through one army and move fast to engage the next served as a huge force multiplier.

This became problematic, however, when the Manchus started moving further south from steppe into China, away from more rural regions in Liaodong Province and running up against walled cities. All the mobility in the world didn't do them much good against a huge stone wall, which their arrows couldn't hurt. The Manchus has no capability for siege warfare, and they lost a few disastrous battles outside walled Ming cities because they didn't know how to engage an opponent behind walls and the defenders' cannons tore them apart.

The changing factor here was the gunpowder weapons the Ming Chinese carried and used. The Ming learned to cast cannons from Jesuit priests and there were skilled craftsmen among the Ming capable of doing so. When the Manchu started moving south, many Ming military and political leaders (especially in Liaodong Province, where there was a certain amount of shared ethnic identity and a good deal of chafed relations with the Ming court) defected to join the Manchu, and many of those who didn't were pressed into service after they were defeated. From these Ming defectors and captives, the Manchu acquired cannons and the ability to cast cannons, and incorporated them into their war machine. Now their forces were able to attack walled cities, but their cavalry's efficacy is still diminished, and the early battles taught the Ming that sieges tended to be a better bet than trying to engage the Manchu cavalry on the field, so much of the remainder of the conquest was won through siege warfare.

The Qing were the last major conquerors of China. Their dynasty lasted from 1635 to 1912. By the time the dynasty faced the sort of major threats that required them to actually field an army, it was the 19th century. Not only were the Qing fairly thoroughly Sinofied by that point (i.e. having lost a good deal of the steppe culture that produced the horse archers in the first place), but technology had changed. Guns were predominant as the primary weapons of soldiers with longer range, faster loading, and better accuracy. They were fighting wars against the likes of the British Empire and Napoleonic France. By the time they had real need of them again, horse archers just weren't going to cut it, even if they still had the cultural memory to train and produce them in significant numbers.

But yeah, the short answer is guns. Looking at other similar cultures like the Mamluk and Ottoman Turks tells a similar story - just look at pictures like this of a Mamluk from 1810, or the standard image of Berber horsemen. Even in cultures where the military tradition remained strong, the horse archers just became horse riflemen. It wasn't until the 20th century, when horses became obsolete on the battlefield (and even then, tell that to the mujahideen) that troops not all that far a cry from horse archers would be phased out entirely.

As a side note, plate armour did pose a major difficulty for the Turkic horse archers in the Middle East who had to fight Crusader knights, but the long and the short of it was that horses can be shot out from under knights, plate armour has gaps, and when you fire enough arrows, eventually you'll hit something soft. The Crusaders counteracted the Muslim horse archers by essentially using their infantry as human shields/a "marching castle," but since the Muslim war machine wasn't quite as based on horse archers as was the Mongols' or the Manchus', they found ways around that issue eventually as well.

Sources:

• "Did Guns Matter?" by Nicola Di Cosmo in The Qing in World-Historical Time by Lynn Struve, editor

• The Manchu Way: The Eight Banners and Ethnic Identity in Late Imperial China by Mark Elliott

• The Military Collapse of China's Ming Dynasty, 1618-1644 by Kenneth Swope