I always suggest (and use) an insulated cooler for long car trips. It can be cheap styro or more permanent hard plastic. Just drill (or punch) a few air holes - but not too many. Put either smaller plastic containers or sacks in the cooler. You can even put some large water or soda bottles with water in the cooler to provide extra thermal mass so the temp changes are more gradual. Unless it is very hot or cold outside, you can even go into a store or restaurant for a little while if they are in coolers. If they are not in a cooler (or if temps are extreme), then it is best to carry a large canvas shoulder bag and keep the animals in cloth bags so that you can carry them with you into restaurants or other places, and nobody will know there are animals in the canvas bag.
As others have said, extreme temps are your main concern. But with an insulated cooler, those concerns will be minimized.