  Here, we investigate the effective response of three-dimensional spacetime
crystals formed by spherical scatterers under a travelling-wave modulation. We
develop an analytical formalism to homogenize the spacetime crystals that
extends the renowned Clausius-Mossotti formula to time-varying platforms. Our
formalism shows that travelling-wave spacetime crystals can be used to engineer
a wide range of classes of nonreciprocal bianisotropic couplings in the long
wavelength limit. In particular, our theory reveals the possibility of
realizing a purely isotropic Tellegen response in crystals formed by interlaced
sub-lattices of scatterers subjected to different modulation velocities.
Furthermore, we introduce a class of generalized Minkowskian crystals that
displays invariance under arbitrary Lorentz boosts aligned with a fixed spatial
direction. We prove that such systems are formed by pseudo-uniaxial materials
with the principal axis aligned parallel to the modulation velocity. The
electromagnetic response of such generalized Minkowskian crystals is
indistinguishable from that of moving photonic crystals.
