It came days after a 37-year-old man was arrested over a video appearing to show a boy driving a lorry on the M1, also close to Dungannon.
First Minister Arlene Foster branded the videos as "crazy and reckless".
Police have opened an investigation after the latest video emerged over the weekend.
The 36-second clip shows a young girl at the wheel of the vehicle on the A4 near Dungannon. The road leads on to the M1.
The girl at the steering wheel in the video giggles while the man smiles. A man and a younger girl are also seen in the cab.
The person who captured the video cannot be seen in the clip.
None in the video are wearing seat belts.
Mrs Foster appealed for the public to help police with their investigations "and make our roads safe".
"Both these crazy videos have been in my constituency. This is reckless for the people in the HGVs and other road users," she tweeted. "A vehicle like this veering out of control into traffic is unthinkable."
Linzi (38) from Downpatrick said she was "shocked" at the videos, adding that there was no excuse for that type of behaviour after her father died on the A1 in December 2015.
Mr McNally (56) from Laurencetown, Co Down, was a former station commander at Newry Fire Station, who specialised in road traffic accidents in his role as a road traffic collision instructor.
He had retired from the Northern Ireland Fire and Rescue Service 11 months before he lost his life after he attempted to cross the Donaghmore junction just outside Newry.
"He would have been horrified," Ms Magee said. "A lot of the concepts of road safety is recognising that everybody is human and everybody will make mistakes but if we abide by the rules of the road and follow the highway code as much as possible then at least you can think you have done everything possible.
"But that kind of behaviour just shows a disregard for the rules of the road, their safety and everyone else's safety.
"They're not even trying to mitigate any risk. There is no excuse for that kind of behaviour. It's just inexcusable."
Seamus Leheny of Logistics UK said the PSNI must investigate the video, as well as the Department for Infrastructure.
A police spokesperson added: "We are aware of footage which has been shared on social media and are making enquiries."
Last week, a 37-year-old man was arrested in Cookstown by police investigating a different video that appeared to show a young boy driving a lorry on the M1. He was released on bail pending further enquiries.