I haven't read about anything like marching fire in Civil War accounts, but Patton in War As I Knew It is emphatic about having all weapons firing at the objective during an assault. In a way he parallels what SLA Marshall said about soldiers not firing their weapons, even though what Marshall wrote was exaggerated. Had he lived longer I'm pretty sure Patton was enough of a master of his craft to have seen when suppressive fire reaches the point of diminishing returns.

This leads us back to the old FM 22-5 tactics drill tactics of the Civil War and First World War that nobody has ever been able to answer -- how do junior officers and NCOs re-establish fire control after the initial bursts of suppressive fire? What next after that? Find a flank? Call in artillery?