  Both applied electric currents and magnetization dynamics modify the
Dzyaloshinskii-Moriya interaction (DMI), which we call current-induced DMI
(CIDMI) and dynamical DMI (DDMI), respectively. We report a theory of CIDMI and
DDMI. The inverse of CIDMI consists in charge pumping by a time-dependent
gradient of magnetization $\partial^2 M(r,t)/\partial r\partial t$, while the
inverse of DDMI describes the torque generated by $\partial^2 M(r,t)/\partial
r\partial t$. In noncollinear magnets CIDMI and DDMI depend on the local
magnetization direction. The resulting spatial gradients correspond to torques
that need to be included into the theories of Gilbert damping, gyromagnetism,
and current-induced torques (CITs) in order to satisfy the Onsager reciprocity
relations. CIDMI is related to the modification of orbital magnetism induced by
magnetization dynamics, which we call dynamical orbital magnetism (DOM), and
spatial gradients of DOM contribute to charge pumping. We present applications
of this formalism to the CITs and to the torque-torque correlation in textured
Rashba ferromagnets.
