Police are appealing to motorists who were travelling on the A74(M) on Saturday at around 4.15 pm, heading southbound on the carriageway at junction 20, Eaglesfield.
Officers detected a speeding black BMW and attempted to pursue the car but it sped off up towards junction 21, Kirkpatrick Fleming.
As the black BMW was approaching junction 21, the driver drove onto the hard shoulder and passed a number of vehicles in an undertaking manoeuvre.
The BMW then exited the slip road and made off. There were no further sightings of the car.
Officers are asking any motorists who may have witnessed the incident or who may have captured it on dash cams to hand the footage into police at Lockerbie Police Station.
Anyone with any information should contact Constable Steven Broatch via 101, quoting incident number 2826 on September 2 2017.