The spin-charge coupled dynamics in a thin, magnetized metallic system are investigated. The effective driving force acting on the charge carriers is generated by a dynamical magnetic texture, which can be induced, e. g., by a magnetic material in contact with a normal-metal system. We consider a general inversion-asymmetric substrate/ normal-metal/ magnet structure, which, by specifying the precise nature of each layer, can mimic various experimentally employed setups. Inversion symmetry breaking gives rise to an effective Rashba spinorbit interaction. We derive general spin-charge kinetic equations which show that such spin-orbit interaction, together with anisotropic Elliott-Yafet spin relaxation, yields significant corrections to themagnetization-induced dynamics. In particular, we present a consistent treatment of the spin density and spin current contributions to the equations of motion, inter alia, identifying a term in the effective force which appears due to a spin current polarized parallel to the magnetization. This " inverse-spin-filter" contribution depends markedly on the parameter which describes the anisotropy in spin relaxation. To further highlight the physical meaning of the different contributions, the spin-pumping configuration of typical experimental setups is analyzed in detail. In the two-dimensional limit the buildup of dc voltage is dominated by the spin-galvanic (inverse Edelstein) effect. A measuring scheme that could isolate this contribution is discussed.