  The Casimir force and thermal Casimir force originating from quantum
electromagnetic fluctuations at zero and non-zero temperatures, respectively,
are significant in nano- and microscale systems and are well-understood. Less
understood, however, are the Casimir and thermal Casimir forces in systems
breaking Lorentz reciprocity. In this work, we derive a formalism for thermal
Casimir forces between an arbitrary number of spheres based on fluctuational
electrodynamics and scattering theory without the assumption of Lorentz
reciprocity. We study the total Casimir force in systems of two and three Weyl
semimetal spheres with time-reversal symmetry breaking for different
orientations of the momentum-space separation of Weyl nodes in both thermal
equilibrium and nonequilibrium. In thermal nonequilibrium, we show that a net
thermal Casimir force exists not only along the center-to-center displacements
of the spheres, but also in the transverse direction to it due to thermal
emission with non-zero angular momentum. Different symmetries of the system
drive a variety of dynamics such as global rotations, self-propulsion, and
spinning of the spheres. We also show that the Casimir energy in thermal
equilibrium depends on the orientations of the Weyl node directions in the
spheres and that the lateral Casimir force will act between the spheres even in
thermal equilibrium to relax the system into the minimum energy state without
transferring net energy and momentum to the environment. The developed
framework opens a way for investigating many-body dynamics by Casimir and
thermal Casimir forces among arbitrary number of spheres with arbitrary
dielectric function tensors in both thermal equilibrium and nonequilibrium.
