Federal Labor is playing down blowback from blue-collar workers after it put the construction union into administration, amid allegations of corruption and criminal links.

Up to 50,000 workers marched in Melbourne and thousands more went out in Sydney, Brisbane and other capitals around the nation, despite threats their pay could be docked if they took unprotected action to protest the action.

Workplace Relations Minister Murray Watt, who oversaw the ushering in of laws to put the CFMEU’s construction division into administration, said the broader labour movement supported what the government did.

The vast majority of marchers were CFMEU members supported by “a couple of other construction-related industries, plus other divisions of the CFMEU in the case of the Maritime Union”.

But most unions and their members didn’t attend the rallies, Senator Watt said. 

“That’s because they recognise the damage that has been done to the union movement,” he told ABC radio on Wednesday.

Labor ministers are also seething at the Greens for backing the protests after their MP Max Chandler-Mather spoke at the 4000-strong Brisbane rally and slammed the administration process, saying it was a threat to all unions.

“We know he likes to grandstand, we know he’s never seen a stage he doesn’t want to jump on,” Finance Minister Katy Gallagher told reporters in Canberra.

“He really needs to explain why he thought it was appropriate to speak at that rally, particularly with such serious allegations of violence, intimidation and misogyny.”

Mr Chandler-Mather said the government used “untested allegations to attack an entire union”, setting a dangerous precedent to seize control of a civil movement or other unions.

The MP linked his support to the principle of a fair trial, adding “the Greens strongly oppose any misogyny, intimidation or thuggery and any allegations of wrongdoing”.

“We live in a country that is not trial by media, it’s … fair process with the presumption of innocence,” he told the ABC’s 730 program on Wednesday night.

Union leaders across the country have chastised Labor and branded the party, which has its roots in the labour movement, as traitors.

The Electrical Trades Union is set to withhold $1 million from Labor ahead of the federal election and use the funds to back a legal challenge against the administration.

Senator Watt said the laws were set up to withstand such a challenge and questioned whether other unions would put their money where their mouth was to back the challenge.

“People make comments in the heat of the moment … let’s wait and see what actually happens and what legal challenges actually get funded,” he said.

“I would encourage them to also reflect on the massive gains that have been delivered to their members by this Labor government.”

Asked about the impact on Labor’s election war chest, Senator Gallagher said it wasn’t about the donations for the party but about “what’s the right response to serious allegations about violence, intimidation and misogyny in a union”.

The Greens’ refusal to back the administration laws or rule out receiving donations from the CFMEU has sparked anger within the government as the minor party targets Labor seats.

Ousted CFMEU national secretary Zach Smith said the union was here to stay and outlined key policies in “the best interests” of members including on Palestine, housing and taxation.

All three issues overlap key tenants of the Greens’ platform.

This post was originally published on Michael West.