We extend the Bethe--Floquet formalism of Joachain and coworkers, originally developed for laser-assisted electron--atom collisions, to inelastic scattering of fast electrons from a many-body condensed-matter target in a time-periodic light field. At first order in the projectile--target interaction, the general Floquet--Fourier cross section separates into exact laser-dressed projectile kernels and a matrix-valued Floquet density response of the target. For spatially structured or translationally non-invariant systems, the latter is the bi-momentum Floquet structure factor; its momentum diagonal defines the Floquet generalization of the dynamic structure factor, constrained by Hermiticity, positivity, and sum rules. As a controlled realization of the general projectile kernel, an eikonal--Volkov approximation for slowly varying inhomogeneous fields yields a finite-momentum-resolution convolution of the target response. In the homogeneous-field dipole limit, the projectile kernel reduces to Bessel-function sidebands and the cross section takes the familiar Bethe--Floquet form. The Floquet density correlator separates exactly into the outer product of the pump-induced coherent mean density and connected fluctuations. In the straight-trajectory nonrecoil limit, the coherent, target-elastic sector defines the weak-coupling PINEM amplitude; repeated coherent insertions generate the PINEM Bessel ladder, with an explicit no-double-counting prescription for combining this channel with connected losses. As worked examples, we evaluate the connected, target-changing cross section analytically for a metal in the Drude and diffusive limits, recovering Bessel-weighted plasmon-loss and diffusive combs, and for a driven two-level model that exhibits off-diagonal Floquet coherences through Bessel-channel interference.