If I take apart my cages (which I usually have to, because the 4' cages will not go in my car) I will put them in a dark box with a dowel stuck through the sides, maybe a towel on the floor, and leave it at that. The chameleon will either fall asleep or sit quietly on the dowel until the box is opened.
I will leave them in their box until their cages are up in the new house, that way they come out and go straight into a proper home.
If they are in a small screen cage and I don't have to take them out I will drape a sheet around the cage instead, to keep it dark. Make sure everything is secure, so no loose branches/vines or hanging plants that can swing around.