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.
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