  The community currently lacks a complete understanding of how resonant
inelastic x-ray scattering (RIXS) experiments probe the electron-phonon
($e$-ph) interaction in solids. For example, most theoretical models of this
process have focused on dispersionless Einstein phonons. Using a recently
developed momentum average (MA) variational approximation for computing RIXS
spectra of band insulators, we examine the influence of both electron and
phonon dispersion in the intermediate state of the scattering process. We find
that the inclusion of either, and their mutual interplay, introduces
significant momentum variations in the RIXS intensity, even for
momentum-independent electron-phonon coupling. The phonon dispersion also
induces nontrivial changes in the excitation line shapes, which can have a
quantitative impact on the data analysis. These results highlight the
considerable challenges of interpreting RIXS data in actual materials.
