  Dynamical magnets can pump spin currents into superconductors. To understand
such a phenomenon, we develop a method utilizing the generalized Usadel
equation to describe time-dependent situations in superconductors in contact
with dynamical ferromagnets. Our proof-of-concept theory is valid when there is
sufficient dephasing at finite temperatures, and when the ferromagnetic
insulators are weakly polarized. We derive the effective equation of motion for
the Keldysh Green's function focusing on a thin film superconductor sandwiched
between two noncollinear ferromagnetic insulators of which one is dynamical. In
turn, we compute the spin currents in the system as a function of the
temperature and the magnetizations' relative orientations. When the induced
Zeeman splitting is weak, we find that the spin accumulation in the
superconducting state is smaller than in the normal states due to the lack of
quasiparticle states inside the gap. This feature gives a lower backflow spin
current from the superconductor as compared to a normal metal. Furthermore, in
superconductors, we find that the ratio between the backflow spin current in
the parallel and anti-parallel magnetization configuration depends strongly on
temperature, in contrast to the constant ratio in normal metals.
