Wisconsin's governor declared a state of emergency Tuesday after some protesters vandalised businesses and set dozens of buildings on fire in a city where police shot a Black man multiple times, apparently in the back in view of his children.

The shooting of Jacob Blake on Sunday in Kenosha was captured on cellphone video and ignited new protests over racial injustice in several cities, just three months after the death of George Floyd at the hands of Minneapolis police touched off a wider reckoning on race.

Blake's father, also named Jacob Blake, told the Chicago Sun-Times that his son is paralysed from the waist down and has "eight holes" in his body. The elder Blake said he didn't know if his son's paralysis was permanent.

Wsconsin Gov. Tony Evers called for protesters to be peaceful, while also saying the National Guard presence would be doubled from 125 to 250 in Kenosha after crowds destroyed dozens of buildings and set more than 30 fires on Monday night.

"We cannot allow the cycle of systemic racism and injustice to continue,” said Evers, who is facing mounting pressure from Republicans over his handling of the unrest that has followed the shooting.

“We also cannot continue going down this path of damage and destruction.”

Blake's father said that he learned Sunday night that officers had shot his 29-year-old son and shortly thereafter he watched the video, which appears to show his son being shot in the back.

“I want to put my hand on my son's cheek and kiss him on his forehead, and then I'll be OK,” the father, who was travelling from North Carolina to Milwaukee, where his son is being treated, told the newspaper.

“I'll kiss him with my mask. The first thing I want to do is touch my son.”

Three of the younger Blake's sons — aged 3, 5 and 8 — were in the car at the time of the shooting, according to Ben Crump, a civil rights attorney representing the family.

Crump scheduled a Tuesday afternoon news conference in Kenosha with Blake's family members to address the shooting.

Police in Kenosha have released almost no information about the shooting, beyond saying the officers were responding to a domestic dispute when they encountered Blake.

The Wisconsin Department of Justice is leading the investigation, which is expected to take several weeks. The officers were placed on administrative leave, which is standard practice in such cases.

The father has not returned multiple messages left by The Associated Press. He is slated to speak at a March on Washington commemoration on Friday organised by the Rev. Al Sharpton. His father and the victim's grandfather, Jacob Blake Sr., was a prominent minister and civil rights leader in the Chicago area in the 1960s.

The man who said he made the cellphone video of the shooting, 22-year-old Raysean White, said he saw Blake scuffling with three officers and heard them yell, “Drop the knife! Drop the knife!” before the gunfire erupted. He said he didn't see a knife in Blake's hands.

In the footage, Blake walks from the sidewalk around the front of his SUV to his driver-side door as officers follow him with their guns drawn and shout at him. As Blake opens the door and leans into the SUV, an officer grabs his shirt from behind and opens fire while Blake has his back turned. Seven shots can be heard, though it isn't clear how many struck Blake or how many officers fired.

Since the shooting, anger has spilled into the streets of Kenosha and other cities, including Los Angeles, Wisconsin''s capital of Madison and in Minneapolis, the epicenter of the Black Lives Matter movement this summer following Floyd's death.

Hundreds of protesters defied an 8 p.m. curfew Monday night, massing in downtown Kenosha, where they were met by a wall of law enforcement officers, including 125 members of the Wisconsin National Guard.

Some clashed with officers and vandalized businesses. There were 34 fires associated with the unrest, with 30 businesses destroyed or damaged along with an unknown number of residences, Kenosha Fire Chief Charles Leipzig told the Kenosha News.

One of the buildings destroyed was the Wisconsin Department of Corrections' probation and parole office. Most staff had already transitioned to working remotely, and all essential operations are continuing, said department spokeswoman Anna Neal.

A city block was cordoned off Tuesday so officials could survey damage. Smoke filled the air and visibility was low as firefighters used water cannons on still smoldering buildings.

“Nobody deserves this,” said Pat Oertle, owner of Computer Adventure. Computers were stolen, and the store was “destroyed,” she said.

“This accomplishes nothing," Oertle said. "This is not justice that they're looking for.”

US Sen. Ron Johnson and US Rep. Bryan Steil, both Republicans, called on the governor to do more to quell the unrest. Steil said he would request federal assistance if necessary.