  We derive the effective interactions between local magnetic moments which are
mediated by Weyl electrons in magnetic topological semimetals. The resulting
spin dynamics is governed by the induced Heisenberg, Kitaev and
Dzyaloshinskii-Moriya (DM) interactions with extended range and oscillatory
dependence on the distance between the spins. These interactions are realized
in multiple competing channels shaped by the multitude of Weyl nodes in the
electron spectrum. Microscopic spins need to be spatially modulated with a
channel-dependent wavevector in order to take advantage of the interactions.
The DM vector is parallel to the displacement between the two interacting
spins, and requires the presence of Weyl electron Fermi surfaces. We also
derive the Weyl-induced chiral three-spin interaction in the presence of an
external magnetic field. This interaction has an extended range as well, and
acts upon the spatially modulated spins in various channels. Its tendency is to
produce a skyrmion lattice or a chiral spin liquid which exhibits topological
Hall effect. Ultimately, the theory developed here addresses magnetic dynamics
in relativistic metals even when chiral magnetism is microscopically precluded.
We discuss insights into the ordered state of the magnetic Weyl semimetal
NdAlSi.
