By K-equivalent of two smooth varieties $X,Y$, we mean there exist a smooth variety $Z$, and birational morphism $q: Z \to X,\quad p: Z \to Y$ , such that $q^* \omega_X \cong p^* \omega_Y$.

Suppose $X,Y$ are K-equivalent, if one has another smooth variety $W$, and birational morphism $t : W \to X$, $s: W \to Y$, would it be $t^* \omega_X \cong s^* \omega_Y$? And why?