"Marching Fire" was just about never used in World War I because it was simply not viable. Let's just define terms here for a moment as they are two sides of the same coin:

Marching Fire is when the entire unit (whatever a 'unit' is in the historical context, be it a company in 1812 or a platoon in 1943) moves up as one body and provides suppressive fire. The goal is not accuracy, the goal is to lay down as much fire as possible for the infantry to make a rapid and shocking advance on the enemy to hopefully cause a tidal wave of crushed morale and routing. In the 18th and 19th century this meant men sprinting, shooting, and then reloading while running again. However in the early 20th century, with entrenched machine guns and barbed wire and mortars and grenades, this was frankly not possible with the average infantry weapon being a bolt action rifle that held 5-10 rounds.

Fire and Maneuver is the other side of the coin of how to advance infantry with supressing fire. This is what the overwhelming doctrine was on the outset and at the end of WWII. This is contrasted with Marching Fire, wherein F&M emphasizes a 'unit' breaking into smaller 'squads' and mutually supporting eachother. This is usually demonstrated in what you may know as 'leapfrogging'. So say you're a a 1915 British Unit (the actual name for 15 men) -- a small group of men, say 5-7, would advance while the rest provided covering fire. Once those men were in cover after maneuvering, they would provide suppressive fire as the group behind them came up and passed them slightly. So on and so forth. This is a far more methodical and 'safe' form of advancing and was far more conducive to the small arms available in 1914-1918.

However on the onset of WWII, Marching Fire had its uses again. I know for a fact it showed up in US and British tactical manuals as early as the 1930's, but perhaps earlier, with the introduction of higher capacity magazines and smaller, lighter squad machine guns. Notably with the former though with the introduction of the Browning Automatic Rifle, the BAR. When in WWII the US issued a semi-automatic rifle as their standard issue and gave a significant amount of men submachine guns and such, and squad MG's only got lighter, marching fire began to be recognized for its use. From a theory standpoint it ensured that, unlike F&M, soldiers were always moving forward and as a whole. In other words, it guarantees something happens rather than it turning into a slog fest. From a morale standpoint it is a well established fact of military history that men are more confident and act with more bravery when operating in large groups acting in unison and all delivering fire simultaneously. This was another advantage of Marching Fire.

Ultimately though it obviously had its downsides. Notably it required a significant amount of combined arms support from tanks, emplaced heavy machine guns, and an already battered enemy position. It also required, due to the obvious inaccuracy of the fire being put on the enemy compared to the well aimed, timed shots of F&M, that soldiers enacting this tactic be relatively close to their enemy line -- usually less than 100m. You can notice in the Fury case, the tanks were rolling up rapidly to the enemy line. The infantry had no point in slowly and methodically firing and maneuvering. They advanced just as rapidly, behind the armor, providing whatever suppressing fire they could.

I hope this clears some things up?