  The Doppler shift of the quasiparticle dispersion by charge currents is
responsible for the critical supercurrents in superconductors and instabilities
of the magnetic ground state of metallic ferromagnets. Here we predict an
analogous effect in thin films of magnetic insulators in which microwaves
emitted by a proximity stripline generate coherent chiral spin currents that
cause a Doppler shift in the magnon dispersion. The spin-wave instability is
suppressed by magnon-magnon interactions that limit spin currents to values
close to but below the threshold for the instability. The spin current
limitations by the backaction of magnon currents on the magnetic order should
be considered as design parameters in magnonic devices.
