Fly a kite in figurative usage, try something out to test opinion. This meaning derives ultimately from the historical sense ‘raise money by an accommodation bill’, i.e., raise money on credit, and this sense of testing public opinion of one's creditworthiness gave rise to the current figurative usage. In the US, go fly a kite! means ‘go away!’
fly in the face of be openly at variance with.
fly off the handle lose one's temper suddenly and unexpectedly (figuratively, with reference to the loose head of an axe).
fly the nest leave one's parents' home to set up home elsewhere (literally, of a young bird, leave its nest on becoming able to fly).
See also fly the flag, pigs might fly.