The recommended approach for working as a veterinarian for short periods over two or more Australian jurisdictions is to obtain registration in the first Australian state or territory in which you intend to arrive and commence work. Once registered, veterinarians can generally work in other jurisdictions under the National Recognition of Veterinary Registration (NRVR) arrangements, which can make it relatively straightforward to move between states and undertake short-term locum work while travelling.
A permanent residential address is not necessarily required for registration. Many veterinarians travelling and undertaking locum work use a base address, such as that of a friend or family member, for correspondence and registration purposes. Veterinarians who initially register in one state and later work in another can generally rely on the NRVR arrangements. However, if they decide to reside permanently or semi-permanently in a particular state or territory, they would normally be expected to transfer your registration to that jurisdiction.
For work in states or territories other than that with which you have registered, notification requirements vary between jurisdictions, and it is important to check the requirements of each relevant veterinary board before commencing work. For example, South Australia requires veterinarians practising under national recognition arrangements to notify the Veterinary Surgeons Board of South Australia within 30 days of arriving in the state, see here for more information on SA requirements.