The spatial relaxation of the electrons in weakly ionized plasmas of atomic and molecular gases acted upon by spatially homogeneous electric fields has been the subject of detailed kinetic studies in recent years. In particular, the relaxation behavior of the electron velocity distribution and of relevant macroscopic transport and dissipation properties of the electrons, initiated by different disturbances, have been systematically investigated and its basic mechanisms analyzed. These studies have revealed that, particularly in atomic plasmas, the electrons generally respond to disturbances with weakly damped, spatially periodic relaxation processes that are characterized by large relaxation lengths and pronouncedly nonlocal properties. Currently, the superposition of magnetic fields gets increasing interest [S. Nakamura, P. L. G. Ventzek, and K. Kitamori, J. Appl. Phys. 85, 2534 (1999)] to additionally control or tune those plasmas that are especially used in plasma processing and laser applications by the imposed magnetic force. This evolution has motivated to extend the spatial relaxation studies to plasmas acted upon by crossed electric and magnetic fields. Thus, based on a consequent kinetic treatment, the present article reports on the considerable modifications of the relaxation behavior of the electrons enforced by the additional electron cyclotron motion in the magnetic field.