  This paper presents an analytical framework for the study of scattering and
diffraction phenomena in spacetime-modulated metallic gratings. Using a Lorentz
transformation, it is shown that a particular class of spacetime-modulated
gratings behave effectively as moving media. We take advantage of this property
to derive a closed analytical solution for the wave scattering problem. In
particular, using our formalism it is possible to avoid spacetime Floquet-Bloch
expansions, as the solution of the problem in the original laboratory frame
(grating parameters are periodic in space and time) is directly linked to a
co-moving frame where the metallic grating is time-invariant (grating
parameters are periodic only in space). In this way, we identify a fundamental
connection between moving metallic gratings and spacetime-modulated
metamaterials, and exploit this link to study the nonreciprocal response of the
structure. Some limitations and difficulties of the alternative nonrelativistic
Galilean approach are discussed and the benefits of the Lorentz approach are
highlighted. Finally, some analytical results are presented in order to
validate the formalism. The results include scenarios involving TM(p) and TE(s)
normal and oblique incidence, even beyond the onset of the diffraction regime.
Furthermore, we show how the synthetic Fresnel drag can tailor the
Goos-H\"anchen effect and create a specular point shifted towards the direction
of the synthetic motion, independent of the sign of the incidence angle.
