Yes! Mostly with bows. Many cavalry cultures did it because it was a common way to do a quick raid. They would ride by a settlement and fire arrows at their victims. Often though, they would be doing this to loot possessions; usually livestock. By being on horseback they could stay mobile and lethal and in the case of stealing livestock, they wouldn't have to get off their horse. It was also used to inflict casualties to goad the enemy into leaving defensive positions or to spur insult to cause an attack. They were a real hardship to people who shared borders with horse archer cultures.

When Native Americans acquired horses some tribes would raid this way, by simply riding by and loosing arrows to inflict casualties.

Eurasian Steppe nomads throughout history did this, like ancient Scythians, Turkic tribes, Magyars, Huns, Mongols, and ancient Massagetae (who killed the Persian Emperor Cyrus the Great in battle!). Horses were their way of life, without a horse a man was often considered impoverished.

I cannot answer for carriage drive bye shootings though, lol.

Erik Hildinger - Warriors of the Steppe: A Military History of Central Asia, 500 B.C. to 1700 A.D

Antony Karasulas Mounted Archers of the Steppe 600BC - 1300AD