Thermoelectric manipulation of the magnetization of a magnetic layered stack in which a low-Curie temperature magnet is sandwiched between two strong magnets (exchange spring device) is considered. Controllable Joule heating produced by a current flowing in the plane of the magnetic stack (CIP configuration) induces a spatial magnetic and thermal structure along the current flow - a magnetothermal-electric domain (soliton). We show that such a structure can experience oscillatory in time dynamics if the magnetic stack is incorporated into an electric circuit in series with an inductor. The excitation of these magnetothermionic oscillations follow the scenario either of "soft" of "hard" instability: in the latter case oscillations arise if the initial perturbation is large enough. The frequency of the temporal oscillations is of the order of 10<^>5 - 10<^>7 s<^>-1 for current densities j ~ 10<^>6 - 10<^>7 A/cm<^>2.