In this chapter we discuss issues specific to applying moment methods with spatially inhomogeneous systems. In particular, we focus on the spatial transport of moment sets by advection, diffusion, and free transport. In Chapter 7, issues related to transport in phase space are thoroughly treated and, here, we will discuss such terms only inasmuch as they affect spatial transport. In Section 8.1, the principal modeling issues that arise with spatially inhomogeneous systems are briefly reviewed. In the sections that follow, we discuss separately moment methods for (i) the inhomogeneous population-balance equation (PBE) (i.e. where the internal coordinates do not include or affect the velocity) in Section 8.3, (ii) the inhomogeneous kinetic equation (KE) (i.e. where the only internal coordinate is velocity) in Section 8.4, and finally (iii) the full inhomogeneous generalized population-balance equation (GPBE) in Section 8.5. Concrete examples, and the corresponding discretized formulas, are provided for each type of system in order for the reader to understand fully the issues that arise when simulating inhomogeneous systems. An important theme running through the entire chapter is the issue of realizable moment sets, and how realizability is affected by spatial transport. Thus, in order to have explicit examples of the numerical issues, we introduce kinetics-based finite-volume methods (KBFVM) for moment sets in Section 8.2. Nevertheless, the reader should keep in mind that these numerical issues are generic to moment transport and will arise with all spatial-discretization methods.