It isn’t specifically Moisture Content that’s important, rather Equilibrium Relative Humidity. I’ll use this term sloppily, for which I apologise, but the ERH is the level of moisture in the surrounding air at which water is neither pushed into nor dragged out of the wood. That’s the ERH of the wood in this closed system.
So in your container full of wood, when the wood has reached equilibrium with the air in the container, the wood will neither gain nor lose moisture vs the air.
If you add the (activated) desiccant to your wood containing vessel and close it, then (presumably) the desiccant will have a much lower ERH. It will drag moisture out of the air. The air is then not in equilibrium with the wood, and will drag moisture out of the wood. Eventually a new equilibrium will be reached. At that equilibrium, there is no reason to assume that the moisture content of the desiccant and the wood will be the same – but the ERH will be. (Definition of equilibrium). It may be that, as it is designed to dry things, the desiccant can have a surprisingly high water content without elevating in ERH that much.
Desiccants are activated by heating them in an oven to drive the air off. When newly activated they have a very low ERH, though this does rise somewhat (as you would expect) when they absorb water.