In systems combining type-II superconductivity and magnetism the nonstationary magnetic field of moving Abrikosov vortices may excite spin waves, or magnons. This effect leads to the appearance of an additional damping force acting on the vortices. By solving the London and Landau-Lifshitz-Gilbert equations we calculate the magnetic-moment-induced force acting on vortices in ferromagnetic superconductors and superconductor-ferromagnet superlattices. If the vortices are driven by a dc force, magnon generation due to the Cherenkov resonance starts as the vortex velocity exceeds some threshold value. For an ideal vortex lattice this leads to an anisotropic contribution to the resistivity and to the appearance of resonance peaks on the current-voltage characteristics. For a disordered vortex array the current will exhibit a steplike increase at some critical voltage. If the vortices are driven by an ac force with a frequency ω, the interaction with magnetic moments will lead to a frequency-dependent magnetic contribution ηM to the vortex viscosity. If ω is below the ferromagnetic resonance frequency ωF, vortices acquire additional inertia. For ω >ωF dissipation is enhanced due to magnon generation. The viscosity ηM can be extracted from the surface impedance of the ferromagnetic superconductor. Estimates of the magnetic force acting on vortices for the U-based ferromagnetic superconductors and cuprate-manganite superlattices are given.