During WWII, it was relatively common for American pilots to fly different types of aircraft in combat after completing conversion training. The 4th Fighter Group (previously Nos. 71, 121, and 133 Squadrons RAF) first flew Spitfires, switching to P-47s in April 1943 and P-51s in March 1944. When large numbers of P-51s first became available to the Eighth Air Force in the spring and summer of 1944, the vast majority of fighter groups were either flying the P-38 or P-47. They proceeded to switch over, and more groups equipped with P-51s from the outset arrived. Only the 56th Fighter Group retained the P-47 until the end of the war, and only the 479th Fighter Group retained the P-38, the other three P-38 groups (20th, 55th, 364th) having converted when P-51s became available. The story of the 354th Fighter Group is an odd one; they were the first group using P-51s to go operational in the ETO (December 1943) and retained them until November 1944, when they received P-47s (like the Eighth and the P-51, the Ninth Air Force, which the 354th operated under, had decided to standardize on P-47s). They got their P-51s back in February 1945.

Pilots who finished their combat tour could request performing a second one; American author and poet Bert Stiles was originally a B-17 crewman, and after completing his tour, he requested a transfer to a fighter group. He was killed in action flying a P-51 on November 26 28, 1944.

Note that the characters from that movie are Army pilots, but I can hopefully provide some illumination on some general concurrent trends. For the US Navy in the first year of the war it was not common but far from unheard of, in part because of the way flight training was set up at the time. In the prewar Navy Student Naval Aviators would check out and qualify on all 3 types of carrier aircraft(fighter, dive bomber, torpedo bomber) before being assigned one to start. And many senior pilots in command of squadrons had spent time in float plane units due to the slim pickings for assignments in the interwar period. This cumbersome time consuming system was abandoned as war became imminent and pilots would be "typed" earlier in the process to go down the defined training pipeline.

That meant that essentially all the naval aviators in the carrier squadrons on December 7th had at least flown the other aircraft aboard. And there were a few instances where it had a real impact, one being at Midway. LCDR Wade McClusky was the CAG on the Enterprise during the battle but was new to the role having taken over less than 2 months prior, having been the commander of the fighter squadron VF-6 aboard previously. What that meant though was that he swapped from flying an F4F Wildcat to and SBD Dauntless, his relative newness to dive bombing also played a role during the battle when he made a mess of the division of his 2 squadrons to attack the Kido Butai. It was only quick thinking by LT Dick Best that prevented both squadrons from completely ganging up on Kaga and leaving Akagi in the clear. Best and his 2 wingmen though scored 1.5 hits out of the 3 of them with best coolly placing a 1,000lb bomb square in the center of her flight deck, dooming the flagship.

In the aftermath of Midway we see further cross type shuffling. Many young fighter pilots suddenly found themselves taken from their squadrons and used to rebuild the gutted torpedo bomber squadrons in the run up to the Guadalcanal landings. This was understandably a less than happy change for many who were frustrated at giving up their role as the aggressive hunter of the skies for a lumbering tub of a bomber. This was also mostly forced on recently arrived pilots fresh from their barely sufficient training stateside, and who were stuck ashore in Hawaii without planes to fly or a carrier to train with for much of the time, so it wasn't really breaking apart experienced squadrons but drawing replacements of little experience from wherever could be found.

The First Team and The First Team at Guadalcanal by Lundstrom offer a great personal look at the US Navy carrier air groups through 1942 when much of the moves were forced out of necessity.

From fighter to bomber is rare, but because of how often new aircraft were developed and put into service, it was very common to see a pilot change aircrafts several times throughout their career as new aircraft were built. I can only really speak for the allies, as the situation may have been different for Axis powers who were desperate for any manpower they could get by the end of the war.

http://www.militaryfactory.com/aircraft/ww2-aircraft-timeline.asp Is a great resource for seeing just how many aircraft were developed and put into service during this time period. WWII was an arms race one very front, including aeronautically. As new technologies were developed, (Things like new turbine technology, night radar, the advent of dive bombers' effectiveness against ships and tanks, countering new anti-aircraft weaponry, etc) pilots were upgraded into newer and more advanced aircraft as old ones were decommissioned or sent to non-front-line areas.

Very few aircraft designs survived combat and front line operation the whole war without being wholely replaced by a newer airframe. The only ones that come to mind are the Zero, which was extremely good for it's time during the first few years of it's existence, but ended up being used even while obsolete later in the war due to Japan's crippled manufacturing infrastructure, and the PBY-Catalina. The Catalina because it was simply so good at it's job of search and rescue. It took an absurd amount of fire to bring one down, and it was able to search a large area of ocean for downed Allied pilots or survivors of sunk warships in the Pacific.

Anyway, I've gotten a bit off topic. I'm sure, sometimes a pilot or crew member would be transferred from a fighter to a bomber, due to the high casualty rates for bomber crews. He likely wouldn't be transferred as a pilot, but rather as a crewmember. For the most part, though, it would be counterproductive to retrain a fighter pilot to act on a bombing crew, because afterwards you'd still have a vacancy for a fighter pilot that'd need to be filled. Until the end of the war, at least on the Allied side, there were new recruits coming in every day, who could be trained specifically to fill a bomber or fighter pilot role, rather than retraining another pilot for it. Unfortunately, I couldn't find a specific source for number regarding pilots transferring, but from my understanding, it was rare, and only happened in extremely extenuating circumstances.

Sources/Reading Material:

Bomber Pilot: A Memoir of World War II, a fantastic book on the subject of bomber pilots in general. Please read if you have any interest in this subject.

http://www.militaryfactory.com/aircraft/ww2-aircraft-timeline.asp is a fantastic website for learning about aircraft and just how different it would be to pilot a bomber rather than a fighter or vice versa.

Pathfinder Pioneer: The Memoir of a Lead Bomber Pilot in World War II is another absolutely fantastic book on the subject in general.

It was not uncommon for pilots to change aircraft type when circumstances conspired for this, usually because of a change in equipment of the squadron/group, or unit (ie, the pilot is transferred).

But the Pearl Harbor situation is incredibly fanciful, as it requires a change in not just squadrons and groups but from fighters to bombers. To suppose a pilot would have to not just learn the peculiar skills of the Doolittle raid (flying a heavy aircraft off carrier decks, long overwater navigation, survival after ditching), but also learn a new, multiengined bomber with its own systems and a large crew, is totally ludicrous.

Which is why that movie is one of the worst of all time for using history to justify a pile of rubbish. Because, you know, Pearl Harbor wasn't an interesting enough situation without the director of Transformers adding from his incredible imagination.