You may be better off asking this question in one of the aviation or physics subs.

However one of the reasons they built biplanes (and triplanes) in WWI is because you have a much greater wing area which means the wing loading is smaller.

A smaller wing loading means a much lower stall speeds which was useful when your engine is less powerful than that of a small family car today.

It also means that take off and landing speeds are reduced, and crucially for the purpose of this conversation, that maneuverability is increased (they pay off is that you increase drag).

The fact that these aircraft were slow meant they had very tight turning circles.

Thus, because late WWI fighters were slow and had very low wing loading, they were very stunty indeed. In fact, the Pitts Special biplane dominated competitive aerobatics for a very long time.

My main sources for this are neither history not aeronautics but from studying Biology at university.

I therefore present a zoological source which goes into some detail about the morphology of winged critters, I hope is OK with the mods.

Followup: What would be the most accurate resource or visual reference for me to understand what an accurate dogfight looked like?

You could do worse than watch Aces High which, although it didnt use actual WWI fighters, did at least use real biplanes instead of CGI

Edit - The Blue Max similarly uses actual biplanes

Edit 2 - The Wikipedia article on wing loading is actually quite good.

Were WWI dogfights feats of aircraft agility or as slow and lumbering as the aircraft themselves seem to be?

A true dogfight was a case of fighter planes using tactics appropriate to the aircraft in question, very much like happened in WWII, Korea, Vietnam and all the way up to today.

There were planes in WWI that were nimble and light, seeming like they could change direction on a wingtip. The Nieuport 11 and 17, Sopwith Pup, Sopwith Camel and Fokker Dr.I (the infamous Triplane) would fall into this category.

Then there were fighters that less nimble, more reliant on pure grunt and horsepower. They were faster in a straight line and generally faster in a dive, but not as nimble. The SPAD VII and XIII and Fokker D.VII would be examples of this strategy.

Finally there were the all-round planes, neither overly nimble nor overly fast, at least when compared to other planes during the entire war. The Albatros D.III, Pfalz D.III or SE5a would be this.

However, and this is important to remember when it comes to WWI fighter planes, technology was evolving at a ridiculous rate. A plane that was dominant in August of 1916 could be obsolete and replaced by January of 1917... the Albatros D.II is a fine example of this, pushed aside by the Albatros D.III.

So a real dogfight could see planes turnfighting in a phonebooth while others slashed in and out using "boom and zoom" tactics, diving in, taking a shot, then climbing away.

Having said all this, straight-line speeds usually maxed out around 130-150mph, so it's all slow in comparison to what we're used to now. At the time, however? When the Ford Model T had a 25hp engine and could reach 40mph? 130mph was pretty fast.

Mind you, any of the fighters of WWI could outmaneuver a fighter from WWII, just 20 years later... and in many cases carried the similar armament. A Sopwith Camel carried two 7.7mm Vickers machine guns, while the Japanese A5M fighter, which first flew in 1935, carried two Type 89 machine guns, which were licensed Vickers E guns.

So WWI fighter planes were, in general, anything but lumbering. Slower than WWII planes, yes, but not lumbering.