Single-particle tracking experiments have recently found that C. elegans zygotes rely on space-dependent switching diffusivities to form intracellular gradients during cell polarization. The relevant proteins switch between fast-diffusing and slow-diffusing states on timescales that are much shorter than the timescale of diffusion or gradient formation. This manifests in models as a small parameter, allowing an asymptotic analysis of the gradient formation. In this paper we consider how this mechanism of rapidly switching diffusive states interacts with a locally varying periodic microstructure in the cell, incorporated through a second small parameter. We show that an asymptotic analysis based on both small parameters yields different results based on the order of limits taken and suggest an explicit relation between the two parameters for when each type of analysis is appropriate. We further investigate a mean first passage time problem for a diffusing protein to gain insight into the effects of the microstructure on the global environment.