It is difficult to break into the airline industry in Australia - simply due to the size of it over here. Compared to the US it is tiny. We only have 2 major airline groups, Qantas which has a 'mainline' operation, Qantaslink(regional) flying turboprops, and their low lost carrier which is Jetstar. The other is Virgin Australia which has a mainline operation, Virgin Australia Regional which is their turboprop and charter operator, and their low cost carrier Tiger Australia. Qantas has it's HQ in Sydney, Virgin is in Brisbane. Jetstar has it's own OCC which is located in Melbourne as does Tiger Australia. All up there would probably less than 300-400 people in Australia that work in an airline OCC, so the jobs are always highly in demand. Both carriers do take external applicants, but it sounds very similar to the US that they do promote mostly from within. From a Dispatch point of view, the position in Australia is not like the USA/Canada. There is no 'license', we do similar duties but there is no 50/50 like over there, dispatchers here prepare the flight plan, flight follow (some flights), but the Captain has overall responsibility to the aircraft. Qantas and Virgin Australia are the only companies with 'Dispatch' departments, Jetstar only provides a flight following service to it's EDTO flights, as it's plans are done overseas. All up there would probably be no more than 100 'dispatchers' in Australia.
As the others have said, working over here is difficult unless you have a partner, much the same as it is for Australians to work in the USA sadly. If you have any further questions feel free to ask and I'll help where i can.