In London, protesters knelt for a minute’s silence before chanting “no justice, no peace” and “black lives matter”.
The majority of the day’s protests were peaceful but in the evening there were disturbances outside Downing Street.
BBC home affairs correspondent Tom Symonds said the protest was largely over when missiles and fireworks were thrown at a police line.
Police horses were used to regain control, but one horse galloped ahead and its rider fell to the ground after hitting her head on a traffic light. Her injuries are not life-threatening, the Metropolitan Police said.
Fourteen people were arrested and 10 officers were injured after a smaller group became “angry and intent on violence”, the force added.
In a tweet, Mayor of London Sadiq Khan told the protesters “I stand with you and I share your anger and your pain” and said the small minority of people who became violent “let down this important cause”.
The protests went ahead despite officials advising against mass gatherings due to coronavirus.
Home Secretary Priti Patel said the social distancing advice was “for the safety of all of us”, while Met Police Commissioner Dame Cressida Dick said the protests were “unlawful”.
Thousands of people gathered in the capital, the majority donning face coverings and many with gloves.
Some held signs made reference to coronavirus, including one which read: “There is a virus greater than Covid-19 and it’s called racism”.
Meanwhile, in Northern Ireland, police said they had issued “a significant number” of fines given the dangers of crowds in the pandemic.
Protests began in the US after a video emerged of Mr Floyd, a 46-year-old black man, being arrested by four officers on 25 May in Minneapolis.
Videos showed Mr Floyd, who was unarmed and in handcuffs, dying after a white policeman knelt on his neck for nearly nine minutes. The footage – seen all around the world – showed him gasping that he could not breathe.
An officer has been charged with murder while three of his colleagues stand accused of aiding and abetting the killing. (Courtesy BBC)