Good question, and a two-part answer.
First, it's important to understand that when solid wood moves with humidity changes, it doesn't move evenly. The most movement is normal at right angles to the grain. Movement in the direction of the grain is usually tiny. So in the case of stave construction, where the grain is vertical, the raw shell would tend to expand or contract in diameter because of changes to the width of the staves. It would not change significantly in shell depth, which would affect tuning more directly. So the seating of tge head is really the main issue -- if the shell expands too much, it's conceivable the head wouldn't fit on the drum.
Also, be aware that the amount of movement (as a percentage of dimension) is different from one species to another. Some are minimally affected while other species could run into problems. This is why white oak is used for flooring more than red oak is. With red oak, you may see buckling in summer humidity and gaps between boards in the winter when the air is drier.
That brings us to the second part of the answer: how well the wood is sealed. In the flooring example, only the top surface normally gets a finish. In the case of a well-made drum, not only the exterior gets sealed and finished, but also the inside and even the edges. With stave construction, the end-grain is on the edges, so it's vital that it get sealed well because end grain absorbs moisture more easily. Fully sealed, the drum should only be minimally affected by humidity issues.
Bottom line: If the shell is sealed well enough, you shouldn't encounter much of a problem with wood movement in normal indoor conditions. You might find yourself needing to tweak the tuning when changing locations, but nothing drastic. If there's a substantial change in conditions (e.g. taking the drum from air conditioning to play an outdoor summer gig), allow some time for the drum to acclimate a bit before you adjust tuning.