The transport equation is formulated against a moving background in the presence of external forces using the effective cross section and source formalism. One-dimensional slab and spherical geometries and two-dimensional cylindrical geometry are treated. Material acceleration terms are expanded, effective cross sections are generated, and moving sources are defined. Legendre expansions that can be coupled to standard multigroup cross sections and sources are suggested. Multigroup representations for the acceleration terms are also obtained. Specific applications are presented and contrasted. The treatment is appropriate for Eulerian reference frames.