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Quantum entanglement is a key element for quantum information that can be generated in a double-cavity magnomechanical system that consists of two microwave cavities, a magnon mode, and a vibrational mode. The magnon mode, which describes a collective excitation of spins, is excited by a strong microwave field. In this system, cavity photons and magnons are coupled via magnetic dipole interaction. The magnons and phonons interact via magnetostrictive interaction, while the two microwave cavities can be connected by a superconducting transmission line. By changing the external driving fields on the two cavities to break the symmetry of spatial inversion, we propose a scheme for asymmetric transmission and entanglement. With the use of current experimental parameters for numerical simulation, we believe our results may reveal a new strategy to build quantum resources for noise-tolerant quantum processors and realize chiral networks.