I have this bone layer, and within that I want to have a switch layer. And I want the contents of this switch layer to be bound to the bones the same way the layers outside the switch layer is. However, no matter what I do, they don't.
Select the switch layer. Click on the Bind Layer tool (in the Bone section of the Tools window, looks like a piece of paper with a bone on it) and then click on the bone in the parent layer you want it attached to. That should do the trick.
That is, if you just need to have the layer move with one particular bone in the skeleton. If you wanted different parts of the sublayers to be controlled by different bones, that doesn't happen. Though, if you weren't aware of it you can put bones inside the Switch layer (they're really a specialized form of Bone layer).