The paper describes numerical modeling of the generation and propagation of the fronts of moving deformation autosolitons in a loaded nonlinear strong medium. It presents solving a system of dynamic equations for solid mechanics, using an equation of state written in a relaxation form that takes into account both an overload of the solid medium and subsequent stress relaxation. The structure of a deformation autosoliton front is investigated in detail. It is shown that the front of a deformation autosoliton that is moving in an elastoplastic medium is a shear band (i.e. a narrow zone of intense shearing strain), which is oriented in the direction of maximum shear stress. Consecutive formation of such shear bands can be viewed as deformation autosoliton perturbations propagating along the axis of loading (compression or extension). A fine structure of a deformation autosoliton front is revealed. It is shown that slow autosoliton dynamics is an integral component of any deformation process, including the seismic process, in any solid medium. In contrast to fast autosoliton dynamics (when the velocities of stress waves are equal to the speed of sound), slow deformation autosoliton perturbations propagate at velocities 5–7 orders of magnitude lower than the velocities of sound. Considering the geomedium, it should be noted that slow dynamics plays a significant role in creating deformation patterns of the crust elements.