Green’s Dictionary of Slang

(Aus. gambling) of a racehorse, not being run to win the race it is competing in (an illegal practce) .

1890		Referee (Sydney) 15 Oct. 1/4: The duties of such an official would [...] entail no end of bickering on the part of owners whose horses are, nine times out of ten, out for an airing .
1902		G.E. Boxall Hist. Aus. Bushrangers 151: The Black Bull was a great sporting house and there were always some race horses there, [...] and, as Mr. Guy was an excellent horseman, he frequently took one or other of these horses out for an airing.

see take for a ride under ride n.