No matter how good your show rabbit is, final placement is often determined by condition being the deciding factor. To that end, transporting your animal in the best possible environment is important. The two prime causes of loss in condition when traveling are heat and rough rides.
First, the cooler you can keep you show animals, the better for holding condition. Try not to have severe drafts blowing directly on the animals, but you are not going to get the temperature too low for show rabbits. Plan your trip to avoid travel in the hottest times.
If you are coming a great distance, plan your travel through the hottest areas at the coolest times of day. If you are depending on natural air circulation to cool animals you need to keep stop times at a very minimum. Having a second driver helps to minimize stop time and heat exposure.
Secondly, you need to travel with rabbits on some kind of padding under the carriers. This minimizes the shaking the rabbit will experience and allows them to travel in more comfort. Rabbits bouncing up and down in the back of a pickup or trailer are going to instantly lose flesh condition. Another advantage of using foam rubber pads under the rabbits is that it helps insulate the rabbits from road heat that arise from the underside of the bed. Old foam mattresses work wonderfully for this purpose and can be easily cut to fit your dimensions.