The household - who are struggling to stay in Australia since they fear persecution in Sri Lanka - have been transferred to Christmas Island detention center immediately, attorney Carina Ford stated, talking in Melbourne.
Mr Dutton defended the government's decision to deport Priya, her husband Nadesalingam and their Australian-born daughters Kopika, 4, and Tharunicaa, 2, in the face of criticism from Labor and the Greens.
The Australian understands that since May there has been more activity out of Sri Lanka alone than all other countries combined across 2017-18.
The mother, father and two daughters have been sent to Christmas Island, as a court injunction delays the government's decision to send them home. A supporter named Angela Fredericks told the Sydney Morning Herald that, "They are now in accommodation near the airport in Darwin. They are Australians as far as the community is concerned", he told Seven's Sunrise.
An action in the Federal Circuit Court on Monday afternoon was shifted to combine with Wednesday's Federal Court hearing, which will examine the youngest child's case for Australia's protection.
Technology is also an issue. Priya and Nades were put into separate vans, driven to Gladstone Airport and flown to a detention center in Melbourne.
The minister said Sri Lankan asylum seekers were increasingly risking the sea journey due to deadly terror attacks in the country on Easter, as well as the Australian federal election in May.
It came as Federal Court Judge Mordy Bomberg extended an interim injunction against the deportation until Wednesday, after lawyers argued Tharunicaa had not been assessed for a protection visa.
"Ministerial intervention powers are created to cater to where a person's not met the legal requirement for a visa but because of humanitarian or national interest grounds the government intervenes".
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"The only other thing is that consent has been given to add the minister for home affairs as a party and the secretary of home affairs", she said.
He also said the couple had been warned prior to having children in Australia that they would not be allowed to stay.
Hundreds of people on the steps of the State Library of Victoria chanted "let them stay", before holding a five-minute silent vigil and then singing Twinkle, Twinkle, Little Star.
He said the "threat" out of Sri Lanka was concerning.
"Priya and Nades fled the country under unsafe circumstances - Priya witnessed the death of her fiance who was burnt alive", he said on Saturday.
"But I feel uncomfortable sending back kids who were born in Australia".
This family had been told long before they had children that there was never a prospect of them remaining in Australia. "Is this the same Peter Dutton, and Immigration Minister David Coleman, who personally intervened past year to stop two Au pairs who were to be deported", Mr Jones 2GB listeners.
"What were the Department hoping to achieve by moving this toddler and her family to literally the furthest landmass that could only loosely be described as falling under Australia's territorial control?"