Pore structure and connectivity determine how microstructured materials perform in applications such as catalysis, fluid storage and transport, filtering, or as reactors. We report a model study on silica aerogel using a recently introduced time-of-flight (TOF) magnetic resonance imaging technique to characterize the flow field and elucidate the effects of heterogeneities in the pore structure on gas flow and dispersion with Xe-129 as the gas-phase sensor. The observed chemical shift allows the separate visualization of unrestricted xenon and xenon confined in the pores of the aerogel. The asymmetrical nature of the dispersion pattern alludes to the existence of a stationary and a flow regime in the aerogel. An exchange time constant is determined to characterize the gas transfer between them. As a general methodology, this technique provides new insights into the dynamics of flow in porous media where multiple phases or chemical species may be present.