Yes it can. The most common problem is respiratory infection (RI).
What is your new cage?(make, material?) What is the substrate? What type or heat source? Where is the water bowl in relation to the heat source?
If you are using cypress mulch it will dry out over the next few days and the humidity will drop. You can help it by turning it each day. Some will say bake it in the oven but you want some moisture so I don't see the point of that.
Having a fan in the room not directly blowing on the cages will help move air in the room and lower your humidity a bit.
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