The family of 24-year-old disabled Warlpiri man Kumanjayi White, who died in police custody in May, has launched legal action against the government.

Mr White, from Yuendumu, died on May 27 after being restrained by two plain-clothed officers following an alleged altercation with a security guard at a Coles supermarket in Mparntwe/Alice Springs. He had been living in supported accommodation under state guardianship.

Last month, his family lodged Federal Court documents, according to reports from the ABC, seeking a declaration that the two officers involved "acted unlawfully" in apprehending Mr White and used force that contributed to his death. The claim alleges the force was "unreasonable" — amounting to assault and battery.

Mr White's aunty, Maxine Carlton, told the ABC the family was determined to pursue justice.

"They messed with the wrong family," she said.

"We're going to fight this all the way, no matter what it takes. We want justice for our son."

The claim also seeks damages from the Northern Territory, arguing it is "vicariously liable for their [police officers] wrongful conduct".

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Mr White's death sparked nationwide vigils and calls for an independent investigation — rejected by NT Police and Chief Minister Lia Finocchiaro, who also declined to visit Yuendumu to meet the family. Advocates say the case highlights systemic issues with policing of First Nations people and the criminalisation of disability.

Acting Police Commissioner Martin Dole has previously defended the internal inquiry, claiming an external investigation could compromise "further action being taken".

However similar cases in other states are often referred to independent oversight bodies, such as the Law Enforcement Conduct Commission in New South Wales.

Warlpiri Elder Ned Jampijinpa Hargraves, Mr White's grandfather, said shortly after the death: "We know that he was held down by two police... he lost consciousness and perished. But at the moment we are in the dark about what really happened."

Speaking at Garma earlier this month, he again called for police to release the CCTV footage and for an independent body to investigate Aboriginal deaths in custody.

Mr Hargraves told the ABC this week the family are "sick and tired of being let down," with court filings noting the only information provided so far had come from "two media releases issued by the Northern Territory government" in May.

"So the family has decided to go ahead and sue the police," Mr Hargraves said, arguing they are doing "because we've had enough".

Mr White's cause of death has not yet been determined. Police confirmed the criminal investigation — codenamed Operation Walcrow — is awaiting a forensic pathology report, with samples sent interstate for further expert analysis.

NT Police have not named or stood down the officers involved. National Indigenous Times understands one of them is NT Police prosecutor Steven Haig, who has continued working in Alice Springs since the incident.

Asked why the officers were not suspended, Acting Deputy Commissioner Travis Wurst said this month: "At this time, the way in which the investigation is being constructed, there isn't a reason to stand them down. At this stage, no charges have been laid."

Chief Minister Lia Finocchiaro said while she empathised with the family, the police process had to run its course, telling ABC Radio, "you can't just make up information to make someone feel better".

At a June protest in Naarm, Mr Hargraves spoke of the community's pain.

"We can't go on like this, we can't live like this," he said. "My jaja (grandson) was killed for no good reason."