Not only did eastern countries employ horse archers to provide mobile firepower, their armies relied primarily on horse archers, and not only that but horse archer tactics were ancient by the time European pistolier tactics came along, and they would outlast them as well.
So let's talk about the caracole. It was a tactic developed during the 16th century as cavalry started adopting wheellock pistols and carbines into their arsenal, and it represents a very interesting tactical development definitely born out of a fluid period of experimentation, as the traditional shock tactics of the lancer were being challenged by increased infantry and cavalry firepower. And it seemed to be rather popular for a time, making the german reiter mercenaries famous and widely imitated. However as a doctrinal synthesis later consolidated in the early to mid 17th centuries it would fall out of use. The reason is because of the terribly limited nature of the firepower that this tactic could provide. Firearms were very difficult to use on horseback, even if the wheellock made it possible at all, because reloading was still impossible to achieve and aiming very cumbersome, so smaller pistols and carbines were used and as such soldier had to get very close to the enemy to deliver fire. This was fine at the beginning, but as arquebus and later musketeers increased in numbers and integration to infantry tactics, they could easily outgun the caracole. It was also found that discipline and cohesion was very hard to manage, and horsemen would prefer shooting outside effective range rather than closing in with the enemy, which in the end was what was meant to be done after the enemy formation was weakened. In the end shock tactics prevailed as championed by swedish king Gustavus Adolphus, adapting firearms not for their long range firepower but for their short range lethality, essentially using their pistols as glorified blunt weapons during melee.
In this sense I wouldn't say that the eastern horse archers fit the same role as the caracole, but the other way around, the caracole attempted to fit the same role as the horse archer, and at under this comparison its failure was spectacular, although such a comparison is rather unfair, as the horse archer represented one of the most effective military units in the whole of the Eurasian landmass. In mass combat situations the horse archer had much greater firepower potential than a pistolier could ever hope to have, in terms of range, accuracy and rate of fire, only perhaps being bested in armor penetration.
This is only on the strictly tactical side, but the prevalence of the horse archer was not solely because of their military usefulness, but because the central Asian steppes were home to the heavily militarized nomads whose entire societies almost completely relied on their skills with the horse and the bow, and as such they provided an almost endless source of dangerous, extremely skilled warriors, and when politically unified could bear their tremendous military strength on their sedentary neighbors, as the Mongols of Genghis Khan are most well known to have done, just as several other powers that preceded them, such as the Gokturks, or followed their footsteps, such as the Timurids. In fact, most of the great eastern sedentary powers of the period were descended from steppe nomad themselves or were severely influenced by their tactics, the Ottomans, Safavids, Mughals, Manchus and Muscovites. In short, the horse archer was to the eastern militaries what the knight was for the European militaries, the dominant figure of war. Eastern armies recruited cavalry from every place imaginable, from warrior noblemen, allied warlords, fief-holding provincial units, mercenaries, local militias, irregular troops, plus of course the household cavalry that responded directly to the central government.
The horse archer was perfectly fit for war in this region of the world, in which the vast, inhospitable expanses of the steppe made strategic mobility an asset, in contrast the forested, hilly, fortified, densely populated European theater of war. As such they relied moreso than merely the bow on their tactical mobility, riding smaller horses than the European knight, but sturdier and better able to travel long distances on minimal fodder, and fighting not as heavy cavalry charging against enemy formations in pitched battles, but relying on skirmishing, ambushing, raiding, harassing, flanking, feigning retreats and attacking supply lines. The horse archer was thus not only better at massed archery in the context of battle, it was far more versatile for a wide range of purposes.
The bow was not their only weapon though, it was just one among many, being such proficient warriors they fought not only with bows but with swords, lances, shields, maces, and sometimes firearms too. As such they ranged from the lightest cavalry of the often poorer nomads to the heavier cavalry employed by the sedentary armies which was better able to furnish itself with armor, often of chainmail. So eastern cavalry could work as heavy cavalry as well, just less specialized in shock combat than the iron clad European knight, and this is really why eastern cavalry did not really shoot at pike squares, because pike squares weren't even a thing in eastern armies since dedicated heavy cavalry wasn't a threat as much as the horse archer who could easily pick off these formations from afar.
This is not a sign of superiority in the eastern tactical system over the European one, just a different context. As already stated, horse archer tactics could not be dominant in the unfavorable European terrain, and their distance from the steppe meant that they could not draw on their skill as mercenaries, so they did the next best thing which was to draw on the skill of eastern European light cavalry, which had evolved to fight the nomad and Ottoman threat. The employment by the Habsburgs of hungarians, croats and poles as light calvary led to the spreading of light cavalry tactics in the 18th century across Europe. Notably these troops were not armed with bows, but relied on cold steel with sabres and lances, as well as firearms. These troops thus performed many of the same functions of the true nomadic horse cavalry in a more limited strategic role and without the same ranged capabilities.
As for firearms, this does not mean that they were doomed to irrelevance on horseback, their usage in heavy cavalry tactics as explained above remained relevant throughout the period. But the more relevant usage was by the dragoons, who were mounted infantry, and rode on horse but dismounted to fight, and as such could better combine the mobility of cavalry with the firepower of guns than the caracole formation could. In fact gunpowder infantry in the east was often of this variety as well, owing to the imperative of mobility. During the reforms of the early 18th century to bring the Russian army more in line with western European standards, most of the cavalry units created were dragoons instead of regular heavy cavalry of the west for this reason.
Sources and further reading:
Kenneth Chase - Firearms: A Global History to 1700
Bert S. Hall - Weapons and Warfare in Renaissance Europe: Gunpowder, Technology, and Tactics
Jeremy Black - European Warfare 1453–1815
Edward J. Erickson and Mesut Uyar - A Military History of the Ottomans: From Osman to Ataturk
Frederick W. Kagan, Robin Higham - The Military History of Tsarist Russia