  In magnetic superconductors vortices polarize spins nonuniformly and
repolarize them when moving. At a low spin relaxation rate and at low bias
currents vortices carrying magnetic polarization clouds become polaron-like and
their velocities are determined by the effective drag coefficient which is
significantly bigger than the Bardeen-Stephen (BS) one. As current increases,
vortices release polarization clouds and the velocity as well as the voltage in
the I-V characteristics jump to values corresponding to the BS drag coefficient
at a critical current $J_c$. The nonuniform components of the magnetic field
and magnetization drop as velocity increases resulting in weaker polarization
and {\it discontinuous} dynamic dissociation depinning transition.
Experimentally the jump shows up as a depinning transition and the
corresponding current at the jump is the depinning current. As current
decreases, on the way back, vortices are retrapped by polarization clouds at
the current $J_r<J_c$. As a result, polaronic effect suppresses dissipation and
enhances critical current. Borocarbides (RE)Ni$_2$B$_2$C with a short
penetration length and highly polarizable rare earth spins seem to be optimal
systems for a detailed study of vortex polaron formation by measuring I-V
characteristics. We propose also to use superconductor-magnet multilayer
structure to study polaronic mechanism of pinning with the goal to achieve high
critical currents. The magnetic layers should have large magnetic
susceptibility to enhance the coupling between vortices and magnetization in
magnetic layers while the relaxation of the magnetization should be slow. For
Nb and proper magnet multilayer structure, we estimate the critical current
density $J_c\sim 10^{9}\ \rm{A/m^2}$ at magnetic field $B\approx 1$ T.
