The issue of transporting dogs in cars crops up on a regular basis.
ALL dogs must be suitably restrained in a vehicle so that it doesn’t distract the driver or injure them if the vehicle stops quickly (UK Highway Code).
However, conditions of exemption state that:
The dog must be muzzled and kept on lead in public places at ALL times, including the car and under the control of a person over the age of sixteen
This means that to follow the letter of the law even if the dog is in a harness attached to a seat belt or in a crate, the dog must still be on a lead and under the control of a person over the age of sixteen if the car they are in, is itself in a public place.