We describe, within the framework of quantum electrodynamics, an interaction between a nonresonant hard x-ray pulse and an electronic system in the presence of a temporally periodic laser field driving electron dynamics in this system. We apply Floquet theory to describe the laser-driven electronic system, and then obtain the scattering probability of an arbitrary nonresonant x-ray pulse from such a system employing the density-matrix formalism. We show that the scattering probability can be connected to the time-dependent electron density of the driven electronic system only under certain conditions, in particular, if the bandwidth of the probe x-ray pulse is sufficiently narrow to spectroscopically resolve transitions to different final states. A special focus is laid on application of the theory to laser-driven crystals in a strongly nonperturbative regime. We show how the time-dependent electron density of a crystal can be reconstructed from energy-resolved scattering patterns. This is illustrated by a calculation of a diffraction signal from a driven MgO crystal.