"Get back in the auto before you die", he told them, before screaming at the child that he was placing him under citizen's arrest and manhandling the father.
In the footage, the producer can be seen repeatedly telling the family they had "assaulted" him and he was placing them under "citizen's arrest".
"In fact I want you dead right now".
Mr.Beeley is a respected figure in the BBC and has worked with the likes of David Attenborough in the past.
But when the two started calmly exchanging details, Beeley stepped in and launched a scathing verbal assault on the motorist, as well as his wife Louse, their 11-year-old son and his mother-in-law.
After telling the whole family that they're under citizen's arrest (even the child), he threatens to kill them all again.
According to a woman speaking in the video, Beeley claimed he was an off-duty police officer and later on the woman says to Beeley: "You said you wanted us dead!"
A passer-by then bravely tries to stand up to Beeley, asking why he is getting involved.
While grabbing Gale by the arms and pushing him against the vehicle, Beeley shouts: "You're under a citizen arrest!".
He claimed Mr Beeley flew into a rage when the family were driving on the motorway when a auto from the inside lane attempted to pull into the middle lane - crashing with their vehicle.
On July 22, he was filmed by a horrified family he had allegedly beckoned off the motorway so he could confront the mum about her "dangerous driving".
"I'm not going to quieten down, because this is a citizens arrest", he continues, before the footage ends.
"This auto pulled across into our lane and hit our auto - our whole auto moved and we nearly got knocked into the outside lane", Simon Gale, driver of one of the vehicles, told the Daily Mail.
Mr Gale claims that Beeley, in a fit of rage, grabbed hold of him around the throat and "raised his fist in a threatening manner".
"It was quite a scary thing".
Hampshire Constabulary said they were called to a non-injury road traffic collision on the M27 between junctions 5 and 7 at 11.30am on July 22.
The footage shows the incident after both parties had left the motorway and entered a Sainsbury's vehicle park.
"The incident involved a Mazda 3, a Peugeot 107 and a Skoda Octavia".
Advice was given to both parties and they were told the two assaults would be recorded and filed.
Beeley has been contacted for comment.
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