The use of full lenght long arms would never have occurred but before the 18th century, it would be namely pistols as they would be small and light enough to use on horseback. By the mid to late 18th century, carbine versions of standard infantry muskets (both in forms of cavalry muskets (shorter muskets, for dismounted action) and cavalry carbines (even shorter and look more like very long pistols for mounted combat).

By the Revolutionary and the Napoleonic era, the French would have a specific unit, the Chasseurs a Cheval, trained for firing on horseback with cavalry carbines but there isn't many reports for that being practical in large groups (as they were scouts, so being able to fire on horse is practical for defense and harassment).

In short, yes. The use of small arquebuses, petronels, or hackbuts by cavalry both mounted and dismounted was gaining quite a bit of traction by the early 1500s, around the same time they were growing popular on foot. In Jacopo di Porcia's The preceptes of warre, first written during the early Italian Wars, he says specifically about gunners on horseback: "For no sort of souldyers, is more profytable than they nor yet doth more myschife and hurte. For no man is so well harnaysed, that can be saulfe from them: such a vyolence is in that warlye instrumente." The largest shift began with the growing availability of wheelock and snaphaunce firing mechanisms towards the end of the Italian wars, which made it easier for horsemen to fire and reload while moving without have to worry about tending to their lit match at all times. By the end of the sixteenth century other sorts of cavalry in western Europe had been almost completely replaced by horsemen armed with pistols or arquebuses.

In the sixteenth century, a "musket" referred a large caliber weapon much longer and heavier than the muskets of the 16th century. Its weight, up to 20 lbs or more, meant that the barrel needed to be supported by a forked stick while firing and it could not be used from horseback. An "arquebus" meanwhile came to refer to a much lighter, smaller weapon which could be easily fired without a rest. As the musket slowly became lighter and more common among infantry in the 17th century the arquebus became almost exclusively a cavalry weapon, similar to the carbine.

https://imgur.com/a/sqED0

This is also where we start to see a distinction between mounted arquebusiers, who acted as more of a "true" light cavalry, and dragoons, who were typically infantrymen armed with full or "half" muskets and given cheap horses for transport purposes. In the illustration above you can see that the dragoon wears his musket slung across his back while the arquebusier's strap allows his weapon to hang muzzle-down near the hip for easy access while riding. Another illustration by Wallhausen shows an arquebusier training to fire at different angles while riding. In combat arquebusiers would be trained to use a drill sometimes called a "caracole"; one at a time each rank would gallop forwards 2-3 dozen yards, fire their weapons, then wheel away and retreat to the rear to reload, allowing a constant fire to be maintained. Dragoons almost always fought on foot, but some writers such as John Crusoe recommend that they be taught how to shoot from horseback as well, just in case the need arose.

Though they still tended to play second fiddle to infantry and heavy cuirassers in large, pitched battles, mounted arquebusiers became absolutely essential to early modern armies and were fielded in large numbers during the 16th and 17th centuries. They were used for scouting, harassment, foraging, seizing important choke points or defensive positions (Sir roger williams thought that 100 dismounted arquebusiers on the right terrain could potentially hold off an enemy force 10 times larger), and a whole host of other important military duties.