In the 16th century European armies began using a countermarch drill to deliver volleys: in a deep formation the each rank of arquebusiers would deliver a single volley then retire to the rear to reload while the ranks behind them stepped forward and repeated the process in order to maintain a continuous fire. Supposedly it was inspired by a drill the ancient Romans used.
What are our sources for any Roman drills similar to this, and what was their purpose of the Romans didn't have any muskets?
William Louis, brother of Maurice of Nassau, described the famous countermarch drill in a letter to Maurice; he claimed to have found an evolution in Aelian's Tactics. As far as I can tell, he's referencing the Chorean or Persian countermarch; Aelian lays out three evolutions. There is the Makedonian, Lakedaimonian, and Persian countermarch. In the first two, the unit will end up occupying the ground in front of or behind the space it originally occupied, in addition to changing facing. In the Makedonian countermarch, the file-leader will turn in place, while the file moves away from the direction they will be facing to take their place behind the file leader. In the Lakedaimonian countermarch, they move towards the direction they are changing their facing to, with the file-leader leading his men onto the ground previously behind the unit. In the Persian countermarch, the file leader takes the place of the file closer, his men following behind, until the file closer and leader have exchanged places. William Louis seems to have had an expensive imagination, applying this technique to firearms.
You can check out this translation of Aelian's tactics.