How to safely move my bearded dragon on a long car trip?

You can either leave him in his terrarium, and move it entirely, or get a small transport box or plastic box with a lid (make sure you cut slits into the box or punch holes in it so air can get into it) and put paper towels into the box. That should be sufficient, Make sure your beardie has access to moisture /some food and is in a place in the car that does not get directly cooled off by the AC. He would prefer to be in a fairly temperate climate. If you are staying overnight somewhere, you could leave him in the car in the box) which will be less stressful than moving him around .