Opposition Immigration spokesman Richard Marles has written to Grant Hehir in the aftermath of Saturday's aborted Operation Fortitude in Melbourne's central business district.

Mr Marles said the Australian Border Force had been brought into ridicule after a press release was issued on Friday quoting Victoria and Tasmania regional commander Don Smith saying ABF officials would be positioned around Melbourne's CBD "speaking with any individual we cross paths with".

The government has blamed the incident and protest that followed on a "poorly worded press release" and says it was never the intention of authorities to conduct a visa blitz.

It emerged on Sunday that the media release had been sent to Immigration Minister Peter Dutton's office last week, but his office says no one read it because it was regarded as "routine".

"What is concerning me is the lack of understanding about the legal powers of officials of ABF which these [Mr Smith's] comments appear to betray," Mr Marles says in his letter.

"Accordingly, I would request that you undertake an investigation in relation to the training that has been provided to officials of ABF as part of its inception, specifically as to: a) the powers that ABF officers possess and b) the circumstances in which these powers can be legally exercised."

Mr Marles said it was critical there be a prompt investigation because of the "significant community anxiety" the incident had caused.

Victorian Premier Daniel Andrews said on Sunday he had received text messages from distressed multicultural community leaders for whom Friday's events had brought back "some very stark memories of great tragedy and fear".

Mr Andrews praised the response of the wider community's action to the operation.

"You saw a very, very Victorian response as people literally took to the streets to protest against something that was ill-conceived and not something supported by my government, not something supported by Victoria Police and indeed not something supported by the Victorian community," Mr Andrews said.

Fairfax Media asked Mr Dutton's office whose idea it was for ABF officials to join the Victoria Police-led operation, and what training and powers officials have.

A Department of Immigration and Border Protection spokesman said their role in Operation Fortitude, if it had proceeded, was to be a small one.

He said the ABF routinely provided "low level support" to state and territory operations and "should operations result in doubts over visa compliance by particular individuals, they are referred to ABF officers for compliance checks".

"In this operation, six ABF officers were to assist partner agencies in various locations by conducting background visa checks on individuals only in the event they were referred to us," he said.

Labor and the Greens said it was an "astounding admission" that Mr Dutton's office had not read the material it received, with both accusing the Minister of "incompetence".

"Minister Dutton is showing himself to be a bumbling and incompetent minister who needs to step up or resign from his position," Greens leader Richard Di Natale said.

Mr Marles told a later press conference the Minister needed "to take responsibility and explain to the Australian people what was intended with Operation Fortitude and who made the decision to abandon this operation".

Opposition Leader Bill Shorten said the government was trying to "to "throw some middle-level bureaucrat in a uniform under the bus" rather than take responsibility for Friday's chaos.

But Foreign Minister Julie Bishop told Network Ten's Bolt Report Labor's reaction was "completely and utterly over-the-top", repeating the government's position that the problem had been a "poorly worded press release".

With Richard Willingham