I just recently moved with my chinchillas. I didn't change climates or anything but as long as you are keeping the room they are in the proper temp I cant imagine that would be too big of a deal. When I moved I waited until the last possible second to put them in their carrier, and then I put their carrier somewhere stable in the care where it would not move around at all. My two chinchillas were totally fine for the move, they just cuddled together and slept the whole time (although they are pretty laid back to begin with). Then when I set their cage up in my new apartment I put it exactly how it was at my old house so that it was somewhat familiar for them. They were kind of more quiet and way more skittish than usual for the first couple days but after that they went back to their normal selves. i did notice that they changed their habits a bit...they sleep in different places than they used to and they spend all day on the bottom level of their cage and all night on the top which used to be the opposite but overall they did just fine!
