Slow swift electrons with low self-inertia interact differently with matter and light in comparison with their relativistic counterparts: they are easily recoiled, reflected, and also diffracted form optical gratings and nanostructures. As a consequence, they can be also better manipulated into the desired shape. For example, they get bunched quite fast in interaction with acceleration gratings in presence of an external electromagnetic radiation, a phenomenon which can be desirable in development of superradiant coherent light sources. Here, I examine the spatiotemporal behavior of pulsed electron wave packets at low energies interacting with pulsed light and optical gratings, using a quantummechanical self-consistent numerical toolbox which is introduced here. It will be shown that electron pulses are accelerated very fast in interaction with the near-field of the grating, demanding that a synchronicity condition is met. To prevent the electrons to be transversely deflected from the grating a symmetric double-grating configuration is necessary. It is found that even in this configuration, diffraction due to the interaction of the electron with the standing-wave light inside the gap between the gratings, is a source of defocusing. Moreover, the longitudinal broadening of the electron pulse directly affects the final shape of the electron wave packet due to the occurrence of multiple electronphoton scatterings. These investigations pave the way towards the design of more efficient electrondriven photon sources and accelerators.