
Judging by the context, it seems to mean something similar to 'to fly a kite for' something, ie express support (not to be confused with 'go fly a kite', a euphemism for 'go away'). But 'fly a kite for' isn't an expression I would personally use either, and I don't think it is that common.
Googling just gets lots of pages about literal balloons and kites, rather than metaphorical ones. And there's nothing relevant in my Concise OED or limited reference library – time to invest in some new books perhaps?