Petty criminals who have served their time and perpetrators who have never set foot in prison are among the New Zealanders awaiting deportation after having their visas cancelled, a New Zealand MP visiting Australian detention centres, said.

The Australian prime minister, Malcolm Turnbull, has arrived in Auckland to meet his Kiwi counterpart, John Key, as a cross-Tasman storm on immigration issues intensifies.

Kelvin Davis, the New Zealand Labour party spokesman for justice and corrections, has told Guardian Australia that the federal government’s decision to cancel visas for long-term residents who have been sentenced to one or more years in Australian prisons makes a “mockery” of the special Anzac relationship.

“I think it’s an embarrassment for Australia,” he said. “The harm that’s been done by this policy outweighs any good it does to Australia.”

Eighty New Zealand citizens have been deported since December, when the government introduced changes to the migration act allowing for the automatic cancellation of visas.

Scott Morrison, who was immigration minister at the time the changes came in, but who now holds the job of treasurer, told reporters on Friday that visa holders are in Australia “at the good graces of the Australian people”.

“At the end of the day, if you’re here on a visa and you’ve committed sexual assault, if you’re a gangster, if you’re a bikie gang member, if you’ve engaged in physical assault or murder or something like that, you’ve worn out your welcome in this country,” Morrison said. “I don’t care how long you’ve been here. You’ve worn out your welcome if you’re here on a visa.”

But Davis refuted that classification.

He said New Zealanders who served short jail sentences for petty crimes and those who have been charged with a crime but given suspended sentences, make up the bulk of those awaiting deportation.

“They’re people you’d have a beer with in the pub,” he said. “They’re not the murderers and rapists [they’re made out to be].”

The Australian and New Zealand Greens have lodged a complaint with the Australian Human Rights Commission on the policy.

Australian Greens senator, Sarah Hanson-Young, has labelled it “arbitrary”.

“The automatic nature of these visa cancellations mean that we don’t just have people who have served prison time for serious crimes,” Hanson-Young told reporters on Friday. “We’ve got to get real here. This includes people with traffic offences, who have been caught shoplifting.”

David said that high-risk offenders are not separated from low-risk ones in immigration detention, recalling the story of a young man who had been sexually abused in his childhood being forced to share a room with a convicted paedophile.

“They’re just being thrown in [to immigration detention centres],” he said, describing the visa cancellations as “callous” and showing “a lack of humanity”.

Davis met with families of New Zealand detainees currently awaiting deportation from Sydney’s Villawood facility on Thursday and Friday. On Friday night he flies to Christmas Island to try and make contact with detainees in the notorious immigration detention centre there.

Key has vowed to have a “blunt” and “direct” conversation with Turnbull about the visa cancellation issue, but Davis said that Key was too busy “being Mr Guy” to strongly put forward the case for New Zealand citizens to the Australian prime minister.

Key has previously warned against strong-arming the Australian government.

“The one thing I would say is we need to negotiate with these guys, not put them in an armlock, because if we try to put Malcolm in an armlock, what will end up happening is he will actually have to face the domestic politics of his own people,” Key told Radio NZ late last month.

Turnbull will leave New Zealand, his first overseas destination as prime minister, on Saturday.