The container used for the transport of the animal can be the same as the capture container if it meets certain specifications. See #2. Make sure there are no bowls inside the container that can crush the animal during transport. Make sure the lid is secured since it would be hazardous to you and the animal if an escape occurred while you are driving. Always use a secured container even if the animal appears calm or unconscious.
Try to have the car warmed up before placing the animal inside the car. Animals do not have a concept of cars and may become agitated at the sounds and movement. In order to reduce stress, please keep the noise to a minimum (no music and soft voices) and do not transport children and/or pets at the same time, if possible.