By using relativistic, axisymmetric, ideal MHD, we examine the motion of the baryon/e+/-/ photon fluid that emanates from a stellar-mass compact object/debris-disk system (a common outcome of many progenitor models). We prove that the motion can be described as a frozen pulse, which permits the study of each shell of the pancake-shaped outflow using steady-state equations. The ejected energy flux is dominated by the electromagnetic (Poynting) contribution, but it can also have a nonnegligible e+/-/radiation (thermal fireball) component. We demonstrate, through exact self-similar solutions, that the flow is first thermally and subsequently magnetically accelerated up to equipartition between kinetic and Poynting fluxes, i.e., ~ 50% of the total energy is converted into baryonic kinetic energy. The electromagnetic forces also collimate the flow, reaching a cylindrical structure asymptotically.