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Roughly 700,000 Kiwis live in Australia but most are denied access to a range of services, including unemployment and disability support, or the right to work in public service or defence roles.
Advocates believe the change of government could be a watershed moment for relations and are keenly awaiting the first meeting between Mr Albanese and New Zealand leader Jacinda Ardern on Friday.
"We've been waiting for a Labor-Labour meeting since we started," Joanne Cox, spokeswoman for campaigning group Oz Kiwi, founded in 2013, told AAP.
Key to their hopes are promises made by the Australian Labor and Greens parties prior to the 2022 election, as well as a high-powered friend in the Labor cabinet.
Jim Chalmers, the federal Treasurer, has been nicknamed the "Member for New Zealanders" for his representation of Kiwis in his south Brisbane electorate of Rankin.
"I proudly represent one of the biggest Kiwi communities in the country ... where I live really is the epicentre of the Kiwi community," Dr Chalmers said in a 2017 parliamentary debate.
"They enrich our community and they also strengthen our economy."
During his nine years in parliament, Dr Chalmers has regularly gone into bat for New Zealanders.
He has fought for Kiwis to be able to access HECS-HELP student loans, and decried John Howard-era changes that cut off a pathway to citizenship and "exposed (New Zealanders) to substantially greater risk of poverty and intergenerational immobility".
"This side of the house recognises and understands those challenges faced by our New Zealand brothers and sisters," he said.
Oz Kiwi are emboldened by Labor's election and hope Dr Chalmers will bring Australia to the table for a route towards citizenship for Kiwis, a path Scott Morrison's government was not interested in walking.
"Citizenship is the magic bullet for us, a lot of the rest is tinkering at the edges," Ms Cox said.
Adding to Oz Kiwi's enthusiasm, Ms Ardern has pledged to talk citizenship when the prime ministers meet in Sydney this week.
Asked whether she would raise "access to benefits and voting rights", Ms Ardern was clear.
"It all comes back to the ability to access citizenship and so will we be raising it? Yes," Ms Ardern said in Wellington on Tuesday.
"New Zealanders contribute a significant amount to Australia. They're taxpayers. They are filling skill gaps."
On Wednesday, Ms Ardern downplayed the idea of a quick breakthrough.
"I'm happy to set out the things that we'll be talking about, I do want to leave a little bit of room for us to have that first conversation," she said.
"Lets see where that takes us ... I always go into these things with optimism."
The ability of New Zealanders to obtain Australian citizenship is largely dependent on their date of arrival.
The biggest step-change occurred in 2001, when then-prime minister Mr Howard cut off automatic access for many Kiwis to an Australian passport, along with voting rights and other welfare support.
In 2016, Malcolm Turnbull's government created a new avenue for those already in Australia but the door remains closed for many.
Oz Kiwi point to an item in Labor's national platform they believe lays the groundwork for change.
"Labor believes that there is an inequity for New Zealand citizens living in Australia under the terms of the Trans-Tasman Travel Arrangements (TTTA)," it states.
"Labor will consider the permanent residency status and potential citizenship arrangements for New Zealand citizens living in Australia under the terms of the TTTA."
The Greens, with a key role in the Senate, support an even stronger position - pledging permanent residence on arrival for all New Zealanders.