  Resonant inelastic X-ray scattering (RIXS) is used increasingly for
characterizing low-energy collective excitations in materials. RIXS is a
powerful probe, which often requires sophisticated theoretical descriptions to
interpret the data. In particular, the need for accurate theories describing
the influence of electron-phonon ($e$-p) coupling on RIXS spectra is becoming
timely, as instrument resolution improves and this energy regime is rapidly
becoming accessible. To date, only rather exploratory theoretical work has been
carried out for such problems. We begin to bridge this gap by proposing a
versatile variational approximation for calculating RIXS spectra in weakly
doped materials, for a variety of models with diverse $e$-p couplings. Here, we
illustrate some of its potential by studying the role of electron mobility,
which is completely neglected in the widely used local approximation based on
Lang-Firsov theory. Assuming that the electron-phonon coupling is of the
simplest, Holstein type, we discuss the regimes where the local approximation
fails, and demonstrate that its improper use may grossly \textit{underestimate}
the $e$-p coupling strength.
