Since the proclamation of King Charles III, a few arrests have been made and protesters moved on in London, Scotland and Oxford.
The Metropolitan Police issued a statement following a viral video from Parliament Square in central London, when a barrister who was holding up a blank piece of paper was asked for his details by an officer.
"We're aware of a video online showing an officer speaking with a member of the public outside the Palace of Westminster," Deputy Assistant Commissioner Stuart Cundy said on Monday.
"The public absolutely have a right of protest. We have been making this clear to all officers involved in the extraordinary policing operation currently in place and we will continue do so.
"However, the overwhelming majority of interactions between officers and public at this time have been positive as people have come to the capital to mourn."
Paul Powlesland, 36, a barrister and nature rights activist from Barking in east London, had travelled to the centre of the British capital on Monday afternoon with "a blank piece of paper".
"Why would you ask for my details?" he can be heard asking the officer in a video.
The officer replied, "I wanted to make sure you didn't have bail conditions (inaudible)".
"You said you were going to write stuff on it, that may offend people, around the King. It may offend someone," he said.
Mr Powlesland said the officer told him he risked being arrested if he wrote "not my King" on the paper.
Elsewhere, a 22-year-old woman has been charged in connection with a breach of the peace after being arrested during the Accession Proclamation for the King outside St Giles' Cathedral in Edinburgh on Sunday.
She was released from custody and is due to appear at Edinburgh Sheriff Court at a later date.
Symon Hill, 45, was arrested on suspicion of a public order offence after shouting "Who elected him?" when he came across a public formal reading of the proclamation of the accession for the King in Carfax, Oxford.
Mr Hill, who works part-time at the Peace Pledge Union, a secular pacifist organisation, was later de-arrested.
On Monday afternoon, a 22-year-old man was arrested "in connection with a breach of the peace on the Royal Mile", Police Scotland said.
Footage appeared to show a man heckling the royal procession as it went past.
Meanwhile, a protester bearing a hand-made sign saying "not my king" was ushered away from the Palace of Westminster by police.
The incident happened as the King was due to arrive for his address to MPs and peers in Westminster Hall on Monday morning.
A number of campaign groups have expressed concern at the way officers are policing protests as the new King is declared, with some warning the arrests may be unlawful.
Ruth Smeeth, chief executive of Index on Censorship, said the arrests were "deeply concerning".
"The fundamental right to freedom of expression, including the right to protest, is something to be protected regardless of circumstance," she said.
Silkie Carlo, director of Big Brother Watch, said: "If people are being arrested simply for holding protest placards then it is an affront to democracy and highly likely to be unlawful."
Jodie Beck, policy and campaigns officer at Liberty, said: "Protest is not a gift from the state, it is a fundamental right. Being able to choose what, how, and when we protest is a vital part of a healthy and functioning democracy."
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