The other thing that your child will need to take charge of will be their own car insurance policy. This is especially nice as newer drivers tend to require more expensive policies. If they are moving far away without a car, you may be able to put the car they used in your own name without listing them as a driver on the policy any longer. Alternatively, if they are moving and taking their car, they can go ahead and get their own policy. You will no longer need to be responsible for it and can see about downsizing your coverage. Â