Most Cats and Dogs, with some exceptions, can be brought to Australia when people are migrating.
The cost is however not cheap, even before considering the actual transport costs, air flights. etc. Quarantine for example will be from around $2,000 for the 10 day stay.
Quarantine costs can be seen at: www.agriculture.gov.au/cats-dogs/quarantine-facilities-and-fees
There are many Pet Travel companies that will arrange everything required to bring your pet cat or dog to Australia.
This can include Import paperwork, Inoculations, pre-flight documentation, etc.
In my view, not something that most people could arrange on their own.
Some Australian Pet Travel companies will have association with similar companies in other countries, so that they can handle both sides of the transport requirements. The foreign company having experience and knowledge of their countries export procedures and the Australian company having experience and knowledge of Australian import procedures.
Thank you to those who have sent me a coffee, it is very much appreciated.
Some Visa Applicants might find an Experienced Australian Migration Agent of great assistance. - ai16