Video report by Matthew Taylor.
Mr Floyd died on May 25 last year after Derek Chauvin was seen on video kneeling on the 46-year-old black man’s neck for about nine-and-a-half minutes.
The Black Lives Matter movement took to the streets across the world, with peaceful protests taking place in Dumfries and Carlisle last summer.
The verdict, which was reached by a jury after less than a day of deliberations and read out on Tuesday afternoon US time, triggered celebrations with crowds flooding the streets with banners and cars blaring their horns.
Reacting to the news, Cumbrian magistrate and former High Sheriff, Marcia Reid Fotheringham told ITV Border: "When I got the news that the verdict was in I was surprised. I didn't expect to get a verdict that early and I was hopefully but on tenterhooks about the outcome."
She continued: "It was a complete mixture of emotion - positive, but some worry about what will happen now because, my concern really was, will there be a reaction from those who do not believe he should be found guilty?
"I'm totally hopeful - and not hopefully that things have settled, but hopeful that it could be the beginning of change."
Listen to ITV Border's interview with Marcia Reid Fotheringham, which was a part of our online series Black Voices in Conversation, below:
Since the Black Lives Matter protests last year, a new group has been formed. Anti Racist Cumbria is also actively dealing with the problem in our part of the world. One of its founders says last night's verdict in America will have a lasting impact.
Janett Walker described it as a 'landmark case'. She said: " It's absolutely a step in the right direction but we cannot afford to sit on our laurels, we can't rest, we have to keep going. There is a still a very, very long way to go."
In the Scottish Borders, Aneela McKenna helps businesses and other organisations be more inclusive. She says there's now a new hope for a change in attitudes.
Speaking to ITV Border, she said: "It matters to the people in the south of Scotland, but it matters to everyone everywhere.
"You take that quote from George Floyd's daughter that 'daddy changed the world' - daddy really did change the world because now we're talking about racism, we're speaking up and calling it out.
"We just can't tolerate it anymore and it shows if we take action then we can see justice."