Mitt Romney, a Republican senator from Utah, joined a group of Christian protesters marching towards the White House. He tweeted photos of himself in the procession, along with the caption, “Black Lives Matter.”
Although Mr Romney has been a rare Republican voice of opposition to Mr Trump, he was joined last week by Republican senator Lisa Murkowski, who said criticism of Mr Trump was overdue.
Mr Trump’s tough approach to putting down protests continued to draw exceptional rebukes from top retired military officers, a group normally loath to criticise a civilian leader.
Former Joint Chiefs of Staff chairman Colin Powell joined them on Sunday, saying Mr Trump had “drifted away” from the constitution. Mr Powell, a Republican moderate, said Mr Trump had weakened America’s position around the world and that in November’s presidential election he would support Democrat Joe Biden.
Condoleezza Rice, who succeeded Mr Powell as secretary of state under President George W. Bush, told CBS she would “absolutely” oppose using the military against peaceful protesters. “This isn’t a battlefield,” she said.
The president has ordered National Guard troops to begin withdrawing from the nation’s capital, whose Mayor Muriel Bowser, a Democrat who jousted with Mr Trump over the use of force in her city, told Fox News there had been no arrests on Saturday despite the protests, with thousands moving through the capital’s streets.
Former Republican presidential candidate Mitt Romney marched with protesters on Sunday, breaking with others in his party who supported Mr Trump’s calls for a militarised response.
Acting Homeland Security Secretary Chad Wolf told ABC that Washington had been “a city out of control” and denied a problem of systemic racism among police.
The Trump administration has proposed no specific policy changes in response to the widespread outrage over Floyd’s death.
Members of the Congressional Black Caucus (CBC) said they would introduce legislation in the House of Representatives on Monday to make policing more accountable.
The legislation is expected to make it easier to sue police officers over deadly incidents, to ban the sort of choke holds that led to Floyd’s death, and to establish a national database to record police misconduct.
One member of the caucus, Representative Val Demings of Florida, a former police chief in the city of Orlando, told ABC that “systemic racism is always the ghost in the room”.
“What we have to do as a nation is hold police accountable,” said Demings, who has been mentioned as a possible running mate for presumptive Democratic presidential nominee Joe Biden.
It is unclear what support the proposed reforms might find in the Republican-controlled Senate – or whether Mr Trump might sign such legislation into law.
Some jurisdictions have moved already to embrace reforms – starting with bans on the use of tear gas and rubber bullets against protesters.
New York Mayor Bill de Blasio told reporters on Sunday that he would cut the city’s police budget and shift some funds to youth and social services, local media reported.
Mr Trump seized on the call by some protesters to slash police funding to attack Mr Biden, tweeting without evidence that “not only will Sleepy Joe Biden DEFUND THE POLICE, but he will DEFUND OUR MILITARY!”
The president is scheduled to host a round-table with law enforcement on Monday.
Mr Biden, who has accused Mr Trump of fanning “the flames of hate,” plans to travel to Houston on Monday to visit Floyd’s family. He will also record a message to be read at Floyd’s funeral on Tuesday.