The generation of high-fidelity distributed multi-qubit entanglement is a challenging task for large-scale quantum communication and computational networks1-4. The deterministic entanglement of two remote qubits has recently been demonstrated with both photons5-10 and phonons11. However, the deterministic generation and transmission of multi-qubit entanglement has not been demonstrated, primarily owing to limited state-transfer fidelities. Here we report a quantum network comprising two superconducting quantum nodes connected by a one-metre-long superconducting coaxial cable, where each node includes three interconnected qubits. By directly connecting the cable to one qubit in each node, we transfer quantum states between the nodes with a process fidelity of 0.911 ± 0.008. We also prepare a three-qubit Greenberger-Horne-Zeilinger (GHZ) state12-14 in one node and deterministically transfer this state to the other node, with a transferred-state fidelity of 0.656 ± 0.014. We further use this system to deterministically generate a globally distributed two-node, six-qubit GHZ state with a state fidelity of 0.722 ± 0.021. The GHZ state fidelities are clearly above the threshold of 1/2 for genuine multipartite entanglement15, showing that this architecture can be used to coherently link together multiple superconducting quantum processors, providing a modular approach for building large-scale quantum computers16,17.