Several points here. First, your 16-mile ranges seem a bit exaggerated; I think you are thinking of naval guns mounted on battleships. Those 16-inch monstrosities were impractical on land; the shells alone weigh, literally, a ton, and are served by machinery, not men. Moreover they have, and need, a whole battleship to absorb the recoil. The 42-cm Big Bertha, the largest howitzer regularly deployed on land (I'm not counting one-off specials like the Paris Gun) had a range of about 8 miles, and was moved by railroad, with a six-hour assembly/disassembly procedure. Field artillery like the famous French 75 had an effective range of six miles.

Second, those maximum ranges are for indirect fire. Artillery was also deployed in a direct-fire mode, "point and shoot" as it were, which was much more effective since you didn't need spotters and a whole cumbrous communication apparatus for correcting fire - the gunners could just directly observe whether they were hitting or not. For this kind of support an advance of a few hundred meters may be sufficient to put you, not out of range, but out of direct observation - cresting a hill, for example, or going through a forest.

Third, there is a difference between outrunning your artillery, and outrunning the barrage. The latter is the line at which your artillery is aiming, forming (you hope) a curtain of shells between your advancing troops and the enemy reinforcements. The barrage was laid to a fire plan worked out in advance, and was very difficult to modify, because you had to get a runner back through the No-Man's Land that your guns had just spent six hours shelling. Indeed, if the word you wanted to get to them was "We have taken Objective X ahead of schedule, please shift your fire to Barrage Line Y", then your runner might have to get back through the fire of your own guns! Nontrivial.

Finally, while the Western Front was pretty well stuck for most of the war, that wasn't the case everywhere and at all times. In the initial 1914 fighting, the German Army sometimes advanced twenty miles in a day - you can see where this would make the Big Bertha useless; by the time it is set up, the fighting is way out of range. Likewise the Michael offensive in 1918, and the Hundred Days counteroffensive, moved pretty quickly. And on the Eastern Front advances of a hundred miles were not unknown.

I'm not a historian, nor will this be a conclusive answer. However, a few points should be kept in mind:

While some artillery had a range of 16 miles, not all artillery is or was created equal. Germany entered WWI with both a massive and considerably more advanced army than those of Europe and America. Their guns had a far, far greater range on average than those of, say, the French. Both the French and the US, if memory serves, relied largely on the 75mm French Modele 75, which had an effective range of 7km (around 4.3 miles). The Germans enjoyed the benefit of artillery whose range far exceeded that, though wasn't as easily deployed. For example, there was "Max," a railroad gun whose range was somewhere around 40km (24 miles), the 21cm Mörser 10 with an effective range of 9km (5.5 miles), the Mörser 16 (11km/6 miles), and Big Bertha with her heavy duty payload being able to reach out and tickle your chin at around 12km (7 miles).

With that in mind, artillery deployment for the allies during the employment of the tactics you address gets a bit more complicated, because you would have allied forces advancing behind a barrage to destroy things like barbed wire (not always effective), sweep trenches free of opposition and nullify the advantage of fortified positions in a general sense by (a) destroying them or (b) suppressing their combat effectiveness until infantry was close enough to absorb the losses imposed and still seize them as objectives. All of this had to be done with artillery that had a much shorter effective range than the German artillery batteries, which means your (you being the allied forces) had to keep your artillery from being decimated by their (the Germans) artillery while also advancing to keep pace with the advances of the infantry. In such an equation, the distance of the infantry from the effective limit of the artillery was always rather close.

On top of this, the front lines were absolutely hell in terms of movement. Prior to WWI, there was typically a problem with getting artillery into place when there was a lot of rain, or when snow melted, or when there was just not a lot of hospitable ground to be had in general. But this was largely an issue of getting from one battle to another in a timely fashion in order to seize a tactical advantage and force the opposition to stand down due to what's called restricted warfare. That means you pick a place and set the proverbial table, and inclement conditions were, at the worst, merely an issue of days. The first world war was completely different, with single "battles" lasting weeks, or even months. Rain filled in shell craters, grass was killed by constant movement and then washed away by the same rain. Long story short: mud. At places such as Passchendaele, this combat metric is displayed en force by the fact that engineers would, on a daily basis, have to make wooden "roads" in order for troops to move, only for the opposition's artillery to target and destroy it, also on a daily basis.

As you can guess, moving artillery quickly and efficiently under such conditons, while also avoiding superior range advantages from the Germans, was all but impossible to do on the regular. Mechanized artillery wasn't even a thing, after all- so during a pitched set of combat maneuvers, infantry would advance behind a "wall" of artillery, then the artillery would have to be packed up and moved forward- usually pulled by oxen or horse teams- in unbelievably difficult terrain, then set up, engage with accurate fire (which wasn't always achieved), and repeat the whole process again numerous times. Logically speaking, there's only so many times you can pull that off before you arrive at the point where your infantry is beyond your effective fire range (because if they sit still for too long German artillery will just annihilate them in detail) and you're stuck in both a literal and metaphorical quagmire.

While the allied approach to artillery at the onset of the war (as with other tactics in general) was based on speed and mobility, the sheer firepower of German artillery combined with constant virtual standstills where a "big victory" was gaining a mile combined with terrain that made maneuverability insanely difficult to achieve, the end result of allied artillery largely resulted in "we have small guns that can't move far or fast, while they have big guns that can't move far or fast."

I apologize for my lack of annotations and references. Hopefully my statements make it easier for an actual historian to address the issues at hand, but I've given a decent (if intensely limited) summary of the problems faces by allied artillery.

Edit for clarification: my statements regarding German range superiority aren't meant to denote that Germany routinely pummeled allied artillery deployments with their own. It was a tactical factor they had to keep in mind from very early on in the war, however. So the complication of advancing behind a barrage works in three stages: first, your artillery had to stay out of the range of enemy artillery. Second, your infantry must advance near the effective limits of fire in order to achieve this goal. Third, your artillery must advance before the infantry leaves your original effective firing range because you haven't cleared further than that. So your artillery has to sweep out to its maximum (but accurate) range, then pack up, move forward, unpack, and repeat the process all before troops (who encountered inconsistent resistance and thus did not maintain uniform or predictable speed) got too far forward for you to realistically and effectively repeat the process as many times as it took. It wasn't an issue of total failure in one cycle so much as it was maintaining a careful balance with absolutely insane impediments to doing so. When the balance gets put of whack, the relation of barrage to artillery starts to slowly suffer until eventually the infantry can (a) retreat (not happening), (b) sit and be decimated (happened more often than it should have) or (c) push forward and hope that the artillery figures things out on their end.