There is also concern from National's other coalition partners. Act Party leader David Seymour said he was "flabbergasted" and "quite stunned" at the actions of "our Anzac mates".
"I would have thought they would show a bit more discretion for the one other country in the world that is so much like them."
United Future leader and Internal Affairs Minister Peter Dunne said New Zealand's response so far had been subservient and inadequate.
"In recent years our foreign policy has become too craven and trade-focused and lacking a moral compass. In short, we have become too silent, lest we cause offence," he wrote in his Dunne Speaks newsletter yesterday.
"Relying on quiet words in diplomatic ears; nods and winks; pull-asides; text messages, or whatever, is not the way to conduct foreign policy ... it is time to abandon the chin-dripping subservience we are lapsing into."
Australian Foreign Minister Julie Bishop met New Zealand counterpart Murray McCully and Prime Minister John Key in New York this week.
Afterwards, she said the Australian Government would talk further about its policy of detaining and deporting non-Australian offenders who have served a prison sentence of a year or more. Ms Bishop said she would ask Australian Immigration Minister Peter Dutton to speak to New Zealand Immigration Minister Michael Woodhouse but gave no indication that the policy would be reviewed. That meeting has not been scheduled.
Meanwhile, authorities here will get more details on deportees from Australia after an information-sharing agreement was signed between the two countries.
New Zealand agencies will now receive six months' advance notice of deportations and details such as case histories, gang connections, fingerprints and photographs.
Justice Minister Amy Adams said the next step was to change the law to ensure offenders who arrived here were subject to the same oversight as they would if they had served their prison sentence in New Zealand.