Of course. Without delving into the peculiarities of the actual tactics of the Mongols and the efficiency of the mounted archers, the combination of the cavalry and firearms was indeed used extensively, especially in the Western European countries from mid-16th to late 17th century. Since the adoption of the wheellock and later various forms of flintlocks, firearms became very useful on the horseback, what facilitated the creation of the formation utilizing firearms as the support or even the main armament, leading to the formation of the units known as reiters in the West and North, and re-armament of the light cavalry commonly using bows as the ranged weapons in the Central and Eastern European armies. It is also worth noting that in case of the latter, bows have seen an extensive use in the Polish-Lithuanian armies along firearms well into 17th century, primarily in the hands of light cavalry, commonly known as Cossacks (no relation to the inhabitants of Zaporozhe), cheremisi or petykhortsy. Western cavalry of 16th and 17th century generally stuck to the firearms.

The most characteristic tactic used by such units was generally known under the French name caracole and was usually utilized by the light cavalry armed with hand weapons and pistols, sometimes also cavalry carbines. The simplest form of caracole called for the cavalry unit in loose formation organized in 10-12 ranks. Riders were approaching the enemy so that they were in the effective range of the firearms. Then, the soldiers in the first ranks (usually three, depending on the circumstances) were firing their pistols, making a turn in place, firing the second pistol (sometimes both were fired before turning) and returning to the end of the formation, so the next rank could repeating the same procedure. Thus, it can be said that this frontal form of caracole was a form of a countermarch adapted to the requirements of the cavalry (here is a simple illustration).

There was also a variant of caracole, where the soldiers were firing not by ranks, but rather by columns. This tactics was especially popular in the Central and Eastern Europe, most notably by riders armed with a single carbine rather than pistols. To perform an attack, riders were approaching the target unit and then, the soldiers forming the outermost column (left or right, depending on the direction of the attack) were riding in from of the rest of the unit, so that while riding forward, they had the enemy on one side and their own unit on the other, then they were firing (often while still riding), and then were turning again, riding along the opposite flank of their unit, then along its rear flank and were returning to their original place. In the meantime, another column was repeating the same maneuver (here's an simplistic illustration). Another variation utilised firing upon the enemy by the front ranks of the advancing cavalry followed by a turn and subsequent firing by the soldiers forming the flank turned towards the enemy. Such maneuver was usually utilized only against units being at a disadvantage in melee (such as musketeers), to minimize the chances that enemy suddenly charges an exposed flank of the unit performing such an attack.

These tactics became popular in the cavalry of late 16th and 17th century due to their versatility. Although their tactics were primarily based on the ranged weapons and mobility, if the firing proved successful in disrupting enemy, the cavalry unit could have easily follow up with the charge to engage in hand-to hand combat (this was especially true in case of enemy cavalry and arquebusiers that were generally weak in defensive), a tactic used to great effect by 17th century reiters, especially in the English and Swedish armies.

Additionally, light cavalry regiments were also often used in conjunction with other cavalry units to disrupt the enemy units before the latter were engaged by the charging heavy cavalry. This took a rather characteristic form in Polish-Lithuanian Commonwealth that in late 17th century still utilized the shock effect of the frontal charge in the manner almost unchanged since the Middle Ages. A good example of the latter tactics could have been observed in the battle of Gniew in 1626, during the Polish-Swedish war. During the charge against Swedish musketeers, light cavalry, riding behind the heavy regiment still moving at a canter pace suddenly accelerated to gallop, overtaken heavy cavalry and turned, riding between the heavy cavalry and the enemy. Without stopping, the soldiers in the outermost ranks fired their pistols and moved along, clearing the way for the heavy cavalry. Swedes, preparing for the charge, decided not to engage the interfering unit, but the firing managed to disrupt their ranks enough to prevent them from regaining cohesion before the still advancing heavy cavalry successfully charged them.

So, to sum this up, the combination of firearms and cavalry was utilized extensively in Europe as soon as the technology made it practical, with the following development of the specific units and tactics increasing efficiency of such an idea.