Officers attempted to stop a silver Mercedes in the town’s Queen Street at about 8.40 pm on Thursday
The car subsequently failed to stop and made off in the direction of Gelston. It was later recovered abandoned near to Kirkcudbright Cemetery.
Constable Thomas Patterson said: “We want to hear from anyone who may have seen this vehicle or knows the identity of the driver, or of the passengers. Anyone who has any information which might help with this investigation is asked to call Castle Douglas Police on 101.”