The title is a play on "bushy hair" along with aborigines stereotypically being from "the bush" country.
Bugs pops out in Golden Gate Park and encounters a man whom he initially thinks is a 'bad guy', but who asks Bugs to hold on to his balloons while he ties his shoelaces. Bugs complies, but soon finds himself drifting off into the ocean. After commenting that "something's gotta happen pretty soon", that 'something' is a stork delivering a baby joey to a kangaroo. The joey bears a strong resemblance to Hippety Hopper, a McKimson character. After a mixup in a cloud, where Bugs is switched with the joey, Bugs finds himself in Australia dropped into a kangaroo's pouch.