(In WWII, the U.S made improvements on their airplanes by studying downed zero Japanese planes. They used parts of various downed planes to try to rebuild an intact plane).
There is some truth to this, but the real story is even more fantastic. A Japanese Zero pilot tried to ditch on an ice field in the Aleutians, but the wheels dug in and the plane flipped onto its back, killing him. US ground forces found the Zero nearly intact, swarmed it under a shroud of secrecy, winched it upright, and ... it needed only minor repairs to fly again. They whisked it intact to California, and had test pilots fly it.
In fact, WW2 US fighter plane design (and gov't contract procedures) was either so slow (years, e.g. P-38 Lightning was in design already in 1939) or so rapid that only one plane actually had its design cycle straddle this Zero-testing event. This was Grumman's F6F Hellcat, which was already in design since 1941. The US Navy, being fairly smart, knew that the F4F Wildcat was outdated and almost-obsolete, and they already planned a replacement around the ginormous Pratt & Whitney Double Wasp, which was a game-changer similar to USA finishing an engine tech.
(Sidebar: Every fighter plane design team had the same uber-idea: build the smallest possible airframe around the biggest available engine. 1930s engines, including Zero, produced ~900 hp. The Double Wasp offered 1600 - 2000 hp, which allowed designers to run amok. Hence a who's-who of radial engine fighters all derived from the DW. Vought said: it should turn a ginormous 16' propeller, but how do you get the wheels to reach the ground? Let's gull the wings -- the F4U Corsair. Republic said: in Germany the 109 flies at 35k feet, we need turbochargers to fight that high -- let's make our hull a double-decker, 1 engine + 1 huge turbocharger stacked vertically, this will be a monster plane so we could put 8 x .50s on it and not vibrate to heck: the P-47 Thunderbolt. Boeing said: how we gonna lug 2-4 tons of bombs 1500 miles round-trip at 28k' and 250 knots, let's put four of these on a really thin wing: the B-29 Superfortress. Grumman said: hot diggity that much power, let's give it short deep wings that fold: the F6F Hellcat. Meanwhile, the inline engines had the Rolls Royce Merlin: the Spitfire and North American P-51 Mustang. Engine tech is a big first step, but Ship Designer still wins.)
At this time, the F6F Hellcat was already nigh-done within its test cycle, almost ready for its trial flights. (Incidentally, it set the world record, still unbeaten, for shortest design-to-service time for a combat plane ever, IIRC about 18 months, which is nuts unless you're losing a war.) Grumman's test pilot(s) flew the captured Zero, and compared its flight profile. We knew already that F6F climbed and flew faster straight (that engine), and of course all US planes with armor dove faster since the P-40. By design, it turned tighter (higher Gs). What they gleaned from flying the Zero was ... nimbleness, in that the lag from stick-and-kick to wing-snapover was very very short, which could potentially let a better pilot eke out small agility advantages over any laggier plane. That is, simply turning in a smaller circle doesn't help you if the duel turns into many many S-curves, flicks, bob-and-weaves, etc, and you lag a little bit on every change-in-direction. As I recall (of course it was classified at the time, so details weren't given then), Grumman tweaked the F6F to be less stable, so that it was more finicky to fly straight, but more responsive to small-and-quick adjustments. This idea wasn't new, and USAF has repeatedly revisited it since, e.g. the YF-29 with forward-swept wings hovers on the edge of instability to the point of self-destruction from the stresses to gain a massive edge in nimble response (compared to other jets).
Anyways, those lessons and design mods got baked into the production F6F, and it was the first US fighter plane that was simply superior in every respect to the old Zero: faster level, more maneuverable, climbed faster, dove faster, heavier guns, armor. And by that time, our pilots were better, we had more of them, and our carriers had radar. Of course, Japan built improved Zeros with bigger engines and more cannon, but they never had the population base to replace the hideous losses in trained air crew.
So -- ultimately, it's all about the population 
. As for the other US fighters, they didn't really change much in response to knowing the enemy plane designs (including captured/allied 109s and FW-190s). It was more about maximizing your own design to do what it already does best, and trusting your tactics and numbers to out-man the enemy. We'd already been fighting and surviving with worse planes, so when our planes became equal or better, we just kept doing that.
This doesn't correlate well with the usual 4X game mechanism of stealing enemy tech via analysis (which I absolutely love to do, btw). The real world is kind of ugly that way. Just imagine if you're a bunch of Intel combat troops and you capture an Apple Mac fighter. What do you learn? To be more Mac-like? 
