When transporting, I put mine inside of a dark container which makes the inside of the container pitch black. I use a dark container because if they can see things going on in the car, or other cars passing you or trees going by your window and such as you're driving, that can sometimes upset them and stress them. Typically, when introduced to no light, a chameleon will fall asleep, even if it just woke up a few hours earlier, because in their mind, that means it is night time. Night time = sleep time.
So, I have a dark container, with a few branches mounted up in the bottom of it, like the idea posted here (I used sticks from my yard). I have a soft cloth lining the bottom of the container as well as the sides of the container, to prevent any injury in case of an accidental fall-over. I also place a piece of paper towel in the container, damp (not soaked) with water, to provide a tad bit of humidity when the lid is closed. No heat pads or anything... I like the container to be nice and cool if at all possible, because this too, induces sleep.
So then I place my chameleon in the container, and wait like 20 minutes or so. He always walks around and tries to crawl out for about 5-10 minutes but after that, he is calm. Then, I just slowly pick up the container and take him to wherever it is that I'm going. And
EVERY SINGLE TIME I've done this (about 3-4 times), he was asleep at the end of the car ride when I went to open the lid. Shortest car ride was 20 minutes, but the longest car ride was 3 hours when I had to bring him to my parents' house with me during Christmas because I had nobody to watch over him when I went to see family.
Works great. The container is super dark so he falls asleep pretty fast. The only light getting into the container is the minimal light that might get through the air holes that I've poked into the lid of the container - not enough light to bother him.
So many ways to do it.... I just figured I'd post how I do it myself, and why.... In case anyone wants to do the same thing.