I'm currently an ATC with ASA, having moved to Australia from overseas. At the time I was recruited (as an ab-initio), I was living in Australia as a Permanant Resident. As I recall, all I needed to do was demonstrate that I had the right to live and work in Australia to meet that part of the selection criteria.
I would have thought that if you obtained permanant residency in your own right (rather than being sponsored in some fashion by ASA - assuming they may do such a thing) then you'd be in a stronger position to negotiate your entry payscale and location with ASA as an overseas controller.
Migration information can be found at
Workers - Visas & Immigration