Suspect Anne Sacoolas agreed to a resolution with Dunn's family in the civil case.
Lawyers for alleged killer of 19-year-old motorcyclist issue public statement.
PC Harpers mother asked Anne Sacoolas if she could "begin to imagine if what happened to Andrew and Harry had happened to one of your children?"
Anne Sacoolas returned to the US after the 19-year-old's death last year.
Anne Sacoolas, 42, was charged with causing death by dangerous driving in December, but the US rejected an extradition request.
Boris Johnson admits relationship may be "imbalanced" in response to claims Harry Dunn's alleged killer worked for American security services.
The family has called for a "full public enquiry into who knew what and when".
Lawyers For Harry Dunn And Epstein Victims Call For Exchange Of Road Death Suspect And Prince Andrew
The pair are calling for Anne Sacoolas to return to the UK and the Duke of York – who has not been summoned by police – to face questioning from the FBI in the US.
The 19-year-old was killed when his motorbike crashed into a car outside a US military base in Northamptonshire in August last year.
The Home Office submitted the extradition bid on Friday.
Harry Dunn's Family Criticise Anne Sacoolas' Lawyer For Claiming She 'Co-Operated Fully' With Police Despite Leaving The UK
A spokesperson for the family also said they were taking claims made by the Trump administration "with a pinch of salt".
The 42-year-old, who is married to a US intelligence officer, is accused of driving the car that hit and killed the Northamptonshire teenager.
“She’s just going about doing whatever she would usually do as though nothing’s ever happened."
The 19-year-old was killed in a motorbike crash in Northamptonshire on August 27.
The president said Boris Johnson had personally asked to set up a meeting between Dunn's parents and Anne Sacoolas.
Dunn's mother and father pleaded for Anne Sacoolas to "get on the plane and get back to the UK, just do the right thing."
Suspect Anne Sacoolas said she had been 'devastated' by the 'tragic accident'.