I don't believe it would stress them out at all. Just put their stuff in there, with at least half straight from their original cage so it's scented like them. Also, if you'd feel better about it, after wiping it down you could rub some of their fleece items on the bars to help, but it's not really necessary.
My two girls came with a cage (they had for about a year before I homed them), I had them less than a year when I replaced it (due to some rust spots) with the Madagascar cage. They stayed in a PP cage, their vacation cage temporarily while I waited for the new cage to be delivered. Then they went into the Madagascar cage, no problem and even located in a different room (from spare room to our bedroom). When I homed Ollie, he had two cages, a mesh one, that was a good size for two to three gliders for travel and his home cage, which I didn't get because it was so big and it was planned that he would be in the Madagascar cage with my girls anyway eventually. So after successful introductions, they all three lived in the Madagascar cage, no problem. Then we moved across country a few months later, they all lived in a vacation Glide-Arium for the road trim, and the large screened (mesh) enclosure of Ollie's after we arrived in new location. So another big change all around plus a new enclosure for the girls. They all adjusted so well and quickly. Then after a little over a month, I got them the Large Sturdy HQ cage, which is the current one they still have, I did haul that baby back with me when I moved back across country. So all the new locations, new cages and all, they all always adjusted just fine, as long as they were fed.
The only issues I had was with our relationship, as out of cage time was not available to them for several months. After I figured how I could swing it, months after the move, it still didn't happen as regularly has it once did, and the tent was very different than the bathroom play, they had grown a little skittish by then, and was able to build a little bond back up. Then we moved back (me and the critters) and it took a while again for me to be able to get them out for playtime, but we've been going pretty good now with regular playtime and they're almost back to how they once were with me.