There has been much recent interest in using transition-state theory (TST) to extend the time and length scales accessible to molecular-level simulations of adsorbate transport in microsporous materials. However, the vast majority of this work has been performed on systems at infinite dilution. The objective of this paper is to obtain fundamental rate constants for adsorbate motion at nonzero loadings using multidimensional TST. More specifically, we focus on systems where the adsorption of a molecule is not highly localized in a single site, but rather distributed throughout an uncorrugated cage. We develop a theory in which high-dimensional TST integrals are approximated using exact lower-dimensional information. The evaluation of the resulting integrals is performed with an importance sampling method involving the insertion of a single molecule, thus improving the statistical quality of the results. The theory is applied to the motion of methane and xenon in the zeolite ZK4, where hopping between α cages is the rate-limiting event. Our results show that hopping rates increase with loading in the cage, which is consistent with experimental trends in the diffusivity. Agreement between our theory and corresponding molecular dynamics simulations is excellent.