  Spin and charge pumping induced by a precessing magnetization has been
instrumental to the development of spintronics. Nonetheless, most theoretical
studies so far treat the spin-orbit coupling as a perturbation, which
disregards the competition between exchange and spin-orbit fields. In this
work, based on Keldysh formalism and Wigner expansion, we develop an adiabatic
theory of spin and charge pumping adapted to systems with arbitrary spin-orbit
coupling. We apply this theory to the magnetic Rashba gas and magnetic graphene
cases and discuss the pumped ac and dc current. We show that the pumped current
possesses both intrinsic (Berry curvature-driven) and extrinsic (Fermi surface
breathing-driven) contributions, akin to magnetic damping. In addition, we find
that higher harmonics can be generated under large-angle precession and we
propose a couple of experimental setups where such an effect can be
experimentally observed.
