Stabilizing entanglement between remote qubits is of fundamental interest, and a crucial resource for quantum networks. Most approaches to generate remote entanglement make use of gate and measurement operations to generate entangled states, which then can deteriorate over time due to decoherence. In contrast, a dissipatively stabilizing protocol could relax the system into an entangled steady state that becomes the new ground state of the distributed system. We analyze and extend an existing proposal; there it was shown that continuous driving can stabilize remote entanglement between a pair of qubits connected through a nonreciprocal waveguide. We extend this scheme by coupling each of the qubits to one or more additional qubits in a chain configuration and show that the steady state of the system is formed by independent Bell pairs of remotely entangled qubits, providing an exciting route toward efficient entanglement purification. We analyze our proposal in the context of superconducting qubits and show that experimental realization is within reach. Our results demonstrate a powerful approach toward robust remote entanglement generation for superconducting qubit networks and modular quantum processors.