Let's be clear about the group in question.
The 'flying tigers' were a group of American pilots, mechanics and support staff who had left the US military and were fighting under the flag of the Nationalist (Kuomintang) Chinese government, with the Chinese sun insignia on their airplanes. And they were very well paid for doing so, a pilot made $600-650/month plus $500 for every Japanese plane they shot down - very good money for 1942. They were mercenaries. The AVG was formed with covert assistance by Franklin Roosevelt in early 1941, but it wasn't able to mount missions until December 20, 1941, by which time the US was officially at war with Japan.
This is distinct from the China Air Task Force, later to become the 14th USAAF, which was a division of the US Army, under US government command, fighting out of China during the war. Most of the AVG were drafted into the CATF, when the AVG was officially disbanded in July, 1942. General Chennault tended to refer to the CATF personnel as 'flying tigers', probably as a morale booster.
While it operated, only one Flying Tiger was taken prisoner: Mac McGarry. As he was a mercenary at the time of his capture, he could have been executed on the spot, but the Japanese didn't kill him, and he was able to escape. This low prisoner account resulted from the fairly small number of AVG aircraft actually in operation, and the very short period of time that the AVG was actually in action - about seven months.
When US airpower began to dominate and especially after the B29 raids on Japan had begun, US fliers who came down in Japanese held territory tended to be executed upon capture, more often than not.