The baby, thought to be just a few weeks old, was miraculously rescued unharmed.
It is understood the transporter rolled down a hill without anyone in the cab, before crashing on Irish Street just after 2.30pm on Tuesday.
Eyewitnesses report hearing a loud bang as the lorry smashed into the front of a bar and a dental practice in the town.
Emergency services attended the scene and worked to rescue the mother and the baby, which was being carried in a car seat, from beneath the vehicle.
Both were taken to hospital in Belfast, where their conditions are said to be stable.
Paul Symington, editor of the Down Recorder, told UTV that the incident had left people in the town distressed.
"The baby was retrieved very quickly, but the mother ... It took over half an hour," he explained.
"Eyewitnesses said she was conscious throughout. The Fire Service was here very fast, and the paramedics. They seemed to get her out relatively quickly."
He added: "It appears that the horse transporter had broken down earlier in the day towards the top of the hill and people had been working on it.
"Suddenly it took off. There was nobody in the cab when it careered down the hill onto the footpath between the two buildings.
"It narrowly missed a man, but unfortunately hit the mother and child."