To evaluate a quantum circuit on a quantum processor, one must find a mapping from circuit qubits to processor qubits and plan the instruction execution while satisfying the processor's constraints. This is known as the qubit mapping and routing (QMR) problem. High-quality QMR solutions are key to maximizing the utility of scarce quantum resources and minimizing the probability of logical errors affecting computation. The challenge is that the landscape of quantum processors is incredibly diverse and fast-evolving. Given this diversity, dozens of papers have addressed the QMR problem for different qubit hardware, connectivity constraints, and quantum error correction schemes by a developing a new algorithm for a particular context. We present an alternative approach: automatically generating qubit mapping and routing compilers for arbitrary quantum processors. Though each QMR problem is different, we identify a common core structure—device state machine—that we use to formulate an abstract QMR problem. Our formulation naturally leads to a compact domain-specific language for specifying QMR problems and a powerful parametric algorithm that can be instantiated for any QMR specification. Our thorough evaluation on case studies of important QMR problems shows that generated compilers are competitive with handwritten, specialized compilers in terms of runtime and solution quality.