Computer simulations and a stochastic Fokker-Planck equation based approach are used to compare the single-particle diffusion coefficients of equilibrium hard-sphere fluids exhibiting identical inhomogeneous static structure and governed by either Brownian (i.e., overdamped Langevin) or Newtonian microscopic dynamics. The physics of inhomogeneity is explored via the imposition of one-dimensional sinusoidal density profiles of different wavelengths and amplitudes. When imposed density variations are small in magnitude for distances on the scale of a particle diameter, bulk-like average correlations between local structure and mobility are observed. In contrast, when density variations are significant on that length scale, qualitatively different structure-mobility correlations emerge that are sensitive to the governing microscopic dynamics. Correspondingly, a previously proposed scaling between long-time diffusivities for bulk isotropic fluids of particles exhibiting Brownian versus Newtonian dynamics [Pond et al. Soft Matter 2011, 7, 9859-9862] cannot be generalized to describe the position-dependent behaviors of strongly inhomogeneous fluids. While average diffusivities in the inhomogeneous and homogeneous directions are coupled, their qualitative dependencies on inhomogeneity wavelength are sensitive to the details of the microscopic dynamics. Nonetheless, average diffusivities of the inhomogeneous fluids can be approximately predicted for either type of dynamics based on knowledge of bulk isotropic fluid behavior and how inhomogeneity modifies the distribution of available volume. Analogous predictions for average diffusivities of experimental, inhomogeneous colloidal dispersions (based on known bulk behavior) suggest that they will exhibit qualitatively different trends than those predicted by models governed by overdamped Langevin dynamics that do not account for hydrodynamic interactions.