I assumed there would be cart-friendly roads through the desert, though that was probably a poorly-researched assumption on my part. I don't think I described the wheels in detail, though, so maybe it could be more like a sleigh. Or maybe, as you say, it can move over the sand by magic.

OR perhaps to be part of a caravan you must pay the caravan leader, who has a jinni with the magical ability to harden sand just long enough for carts to travel on. I suspect it would be quite the profitable business.