As others have stated, keep the metal oiled and you'll be fine.
Here in LA (lower Alabama) controlling the moisture content in bare wood is a constant battle. I strive took keep 8% MC (moisture content) in the material before finishing. 7 to 8% is acceptable for hardwoods, with 7% preferred. 12% is fine for softwoods. Keep in mind that heat alone does not change the MC in wood. Heat does, however, cause moisture changes to occur faster. Fortunately, most wood on firearms has at least a seal coat on all surfaces, which greatly reduces and ebbs the flow of moisture both in and out of the wood.
For anyone interested, here's is the basic relationship between wood's MC and RH (relative humidity), with equilibrium moisture content (EMC) shown as well. EMC is what actually affects the MC.
0% RH = 0% MC = 0% EMC
30% RH = 6% MC = 6% EMC
50% RH = 9% MC = 9% EMC
65% RH = 12% MC = 12% EMC
80% RH = 16% MC = 16% EMC
99% RH = 28% MC (approx.) = 28% EMC (approx.)
Most homes with heat and AC will run 6% MC in the winter and 9% in the summer. In most of North America the outside EMC is 12%, summer and winter. Along the coasts and in the SE, 16% EMC is common.
If you haven't figured it out by now, I'm kind of a nerd when it comes to MC in wood, but that's because it affects wood products manufacturing in ways that would amaze you. The bottom line is that so long as you store your firearms in an environment with somewhat stable RH - like a room with regular HVAC -you shouldn't have any problems.