A long-range dynamic interaction between ferromagnetic films separated by normal-metal spacers is reported, which is communicated by nonequilibrium spin currents. It is measured by ferromagnetic resonance and explained by an adiabatic spin-pump theory. In such a resonance the spin-pump mechanism of spatially separated magnetic moments leads to an appreciable increase in the resonant linewidth when the resonance fields are well apart, and results in a dramatic linewidth narrowing when the resonant fields approach each other.