About 10,000 protesters marched on the Michigan state capitol on Tuesday, in a show of force by unions fighting the spread of controversial right-to-work legislation in a growing number of US states.

Protesters defied freezing temperatures to demonstrate against union-limiting legislation passed by the Republican-controlled legislature in Lansing.

Union members had begun massing outside the capitol building before dawn, with the numbers swelling into the thousands as buses delivered opponents of the measures from across the state.

Opponents of the legislation say it will lead to lower wages and fewer protections for workers. Supporters say the package of bills will help create jobs.

The latest bill, passed on Tuesday by the Republican-controlled state House of Representative, deals with public-sector workers. Another bill focusing on the private sector was approved last week. The Republican governor of Michigan, Rick Snyder, is expected to sign them into law by Wednesday.

Hundreds of protesters were in the capitol building as the latest vote was taken. Union members, many wearing red T-shirts, ringed the circular balconies that span four floors inside the elaborate dome of the state capitol. The deafening noise could be heard throughout the building as protesters stamped their feet and clapped their hands.

"Right to work – shut it down," protesters chanted, as police officers, some armed with what appeared to be paintball guns, looked on.

Union members from around the country rally at the Michigan state capitol to protest a vote on right-to-work legislation on Tuesday. Photograph: Bill Pugliano/Getty Images

The capitol was closed briefly at around 10am due to overcrowding, said inspector Gene Adamczyk from the Michigan state police. He said some protesters had since left and officers were allowing people in on a one-in-one-out basis.

"We have over 10,000 between inside the building and outside the building right now," Adamcyzk said. As of 10.30am there had been no arrests or injuries, he added.

Outside the capitol building, thousands of union members fanned out across a frozen lawn. A street leading towards the east of the city was thick with flags and signs as protesters bellowed a mixture of union chants and anti right-to-work slogans.

"I know what happened in other right-to-work states," said Jean Karr, who was stood with her shoulders hunched to try and keep out the cold. "Michigan's always been a union state and it's always been one of the top states."

Karr, a retired nurse who lives in Ann Arbor, Michigan, said the impact of the right-to-work legislation could be severe. "The unions are going down which means the Democrats will go down because that is a big supporter of the Democrats. When you lose the Democrats you lose all social programmes, you lose everything."

Gary Chesnutt, a 59-year-old United Auto Workers member, had come to the rally dressed as Santa. The one giveaway was the blue hat – worn to demonstrate support for the Democrats, Chesnutt explained. "Unfortunately our state capitol is full of the GOP, which are wrong for workers," Chesnutt said. He had travelled from Hillman, in north Michigan, to protest the "right-to-work-for-less bill", Chesnutt said.

"It'll destroy our economy over the long term. It's a blatant out-and-out attempt to break the unions, and I don't like freeloaders. There'll be freeloaders in the workplace getting all the benefits of wages and time off and grievance procedure, for free, that I'll have to pay for in my union. It's plain wrong.

"People have got a right not to go get employed in a job that has a union. They can go anywhere else, there's lots of places that don't have unions."

Michigan has the fifth-highest percentage of union workers in the US, with 17.5% of workers in union jobs. Michigan will become the 24th state to introduce "right to work" legislation when Snyder signs the package of bills into law.

Unions argue that wages in right-to-work states are lower than those which do not have the legislation. In February last year a study by the left-leaning Economic Policy Institute found that wages in right-to-work states are on average 3.2% lower than states without such legislation.

'Right to work – shut it down,' protesters chanted, as police officers looked on. Photograph: Julie S Dermansky/For the Guardian

Opponents of the law in Michigan also say that the bills had been rushed through the legislature, something Republicans deny. Democrats said Republicans had wanted to act to pass the law before a new state senate takes office next month – when the Republicans will be weakened, as the party lost five house seats in the November elections.

In one signal of the hostility between opposing sides, supporters of the right-to-work legislation booked the steps of the state capitol building and pitched two huge tents on the capitol lawn.

As the grass filled with thousands of union demonstrators the two largely empty tents became a subject of ridicule among some members of the crowd.

Annie Patnaude, from Americans for Prosperity, the conservative organisation which had pitched the slightly busier of the two tents – there were around 20 people inside the cavernous space at around 10am – said more supporters of the legislation had wanted to come to the capitol but were intimidated by the union presence.

"On Thursday there were a lot of threats, intimidation and violence. One of our activists even got punched in the face," Patnaude said.

Of the right-to-work law Patnaude said "the basic issue is freedom".

"No one should be forced to join and pay dues to a union in order to keep their job. That's the basic issue."

By 12.30pm both tents where on the ground, reportedly yanked down by protesters. The Americans for Prosperity tent had been trodden into the ground, dirty bootprints all over the white canvas. A group of men wearing hard hats were posing for photos in the middle of the wreckage.