New Zealand citizens living in Australia should be able to earn the same rights as Australian citizens living in New Zealand.

Historical agreements and arrangements between Australia and New Zealand's Trans Tasman Travel Arrangement (TTTA) used to be reciprocal, but now they are vastly different.
New Zealander citizens living in Australia have had their basic rights and entitlements continuously eroded since February 26th 2001.
Australian citizens and Australian Permanent Residents living in New Zealand however, receive full rights, benefits and entitlements without prejudice or discrimination.