Many current problems of interest in quantum non-equilibrium are described by time-local master equations (TLMEs) for the density matrix that are not of the Lindblad form, that is, that are not strictly probability conserving and/or Markovian. Here we describe an generic approach by which the system of interest that obeys the TLME is coupled to an ancilla, such that the dynamics of the combined system-plus-ancilla is Markovian and thus described by a Lindblad equation. This in turn allows us to recover the properties of the original TLME dynamics from a physical unravelling of this associated Lindblad dynamics. We discuss applications of this generic mapping in two areas of current interest. The first one is that of "thermodynamics of trajectories", where non-Lindblad master equations encode the large-deviation properties of the dynamics, and we show that the relevant large-deviation functions (i.e. dynamical free-energies) can be recovered from appropriate observables of the ancilla. The second one is that of quantum filters, where we show tracking a quantum system undergoing a continuous homodyne measurement with another quantum system of the same size will inherently be inefficient in our framework.