There wasn’t any one-to-one reflection: the nature of the war in the air and that on the ground was just too different.
In the most literal sense of “arms race”, the guns mounted on aircraft were more-or-less ground weapons adapted, often very slightly, for air service. For example, in Britain the Lewis light machine gun was flown without the cooling jacket that was ubiquitous on the ground because of the perception that, in the air, it was useless. Lewis guns on the ground, however, still had the jacket right up through the end of the war.
There was little change in the machine guns used from the beginning of the aerial machine gun era in 1915 right through November 1918. The British used the Lewis and the Vickers medium machine gun, the French used a hodge-podge of weapons including the Lewis, the Vickers, and occasionally a Hotchkiss. German guns evolved the most, but were almost entirely based off the MG 08 and its later revisions or redesigns.
There was plenty of inventiveness going into aircraft ordnance but for the most part these innovations, most famously synchronization gear, were meant to solve air-specific problems. A famous example was the French SPAD S.XII, which fired a 37mm cannon through the propellor shaft; the cannon used was bespoke for the aircraft and didn’t reflect what was going on below. The biggest contribution aerial machine guns made to general gunnery was the metal disintegrating-link ammunition belt: traditional fabric belts presented a fouling hazard on an aircraft, the disintegrating link entered RFC service in 1916, and soon it was being copied both on the air and on the ground.
The main aerial air race, you’ll understand, was in terms of improving aircraft. The utility for the ground war was therefore pretty limited. Even the most versatile types like the Rolls-Royce Eagle or the legendary Mercedes D.III series could power aircraft of all shapes and sizes and even airships but weren’t good for much on the ground.
Bear in mind that, in the air, the cutting edge became obsolete very quickly. The Fokker Eindecker scout plane was so terrifying to the British in 1915 they ordered missions formerly conducted by one aircraft to be flown by four: by the end of 1916 even the newest Eindeckers were worse than useless. The British B.E.2 was one of the finest observation aircraft in the world when it was introduced in the beginning of the war, and while it served in various roles until the end it was utterly incapable of meaningful resistance in combat. The most famous single-seater scout of them all, the Sopwith Camel, was introduced in the middle of 1917, becoming out of date (except with the faster Bentley engine) by spring 1918, and by the summer was increasingly a ground support aircraft rather than being risked against German scouts.The only aircraft design that served a couple years with much grace was the Russian Ilya Muromets, a pre-war design and the first “heavy bomber”: it was borderline invulnerable through 1914 and 1915 and still useful until the end of 1916. And it had the advantage of fighting on the Eastern front, which was very much the secondary theatre in aircraft terms.
Compare all that to the French soixante-quinze gun was invented in 1897 and served through 1918; only the early tanks, which were literally being invented during the war, really compare to aircraft in their rapid obsolescence. Aircraft were immature, but many of the arms being fought with on the ground were well-known in principle. This was still an era where one, two, three, or even four wings could be fitted to a potential combat aircraft; nothing was non-negotiable.
Suggested Reading
Bennett, Leon. Gunning for the Red Baron. College Station: Texas A&M University Press, 2006.
pretty much any of the Osprey books on any aircraft you’d be interested in
swing by the Canadian War Museum in Ottawa, ON to see and compare aircraft and ground-based Lewis and Vickers guns.