Police confirmed his heartbroken dad Phillip, 41, was in charge of the vehicle when the accident happened.
Karl, who had earlier been snapped proudly sitting next to his dad, is thought to have fallen from the back of the steam traction engine into the path of the trailer it was pulling.
Paramedics arrived just before 3pm yesterday but Karl was pronounced dead at the scene. A post-mortem exam revealed he died due to a head injury.
Mr Doran was taken to hospital after the incident to be treated for shock.
According to friends, father and son were were both steam enthusiasts and regular volunteers at Beamish, which is also known as the Living Museum of the North.
Visitor Lisa van Dijk wrote on the museum's Facebook page: "I didn't sleep last night. My husband took a photo of me with the little boy and his father in the background only 15 mins before the accident. I can't stop thinking about it."
DCI Victoria Fuller, who is leading the investigation, said: "We are treating this as a tragic incident.
"We need to speak to key people who may have been on a bus or tram which passed the traction engine at the time of the incident around 3pm."