First keep in mind that the climate indoors may be quite different than that of the outdoors, and that indoor climate generally doesn't vary as much as outdoor climate.
You should expect to have some changes in the piano after your move, as the wood adjusts to the drier climate. The piano may go out of tune pretty dramatically. It may require a "pitch raise", and will probably not be really stable until it has been tuned at least two or three times, and spent at least a few months in its new home.
It may be that the soundboard may crack, and while that sounds horrible, it is not necessarily something that you need worry about. That is really only evidence as to the present condition of that piece of wood, being much drier than it was when it was first installed in the piano. The piano may perform without any appreciable difference in tone, even with several cracks in the soundboard.
There is one other area of concern for you, though, and that is in the area of the pin block, which is the piece of wood wherein all the tuning pins are driven. This is a block that is made up of several laminations of maple, many of them quarter-sawn, and cross grained, so to develop a tight grip on the steel tuning pins. If the tuning pins were not particularly tight before, they may get loose enough with the loss of moisture in the wood, to lose the ability to resist the pull of the string (160‑220 lbs per each string/pin).