Source-sink systems are metapopulations of patches with different, and possibly temporally varying, habitat qualities, which are commonly used in ecology to study the fate of spatially extended populations. We propose new techniques that disentangle the respective contributions of demography and dispersal to the dynamics and fate of a single species in a source-sink system. Our approach is valid for a general class of stochastic, individual-based, stepping-stone models, with density-independent demography and dispersal, provided the metapopulation is finite or else enjoys some transitivity property. We provide (1) a simple criterion of persistence, by studying the motion of a single random disperser until it returns to its initial position; (2) a joint characterization of the long-term growth rate and of the asymptotic occupancy frequencies of the ancestral lineage of a random survivor, by using large deviations theory. Both techniques yield formulae decoupling demography and dispersal, and can be adapted to the case of periodic or random environments, where habitat qualities are autocorrelated in space and possibly in time. In this last case, we display examples of coupled time-averaged sinks allowing survival, as was previously known in the absence of demographic stochasticity for fully mixing (Jansen and Yoshimura, 1998) or partially mixing (Evans et al., 2012; Schreiber, 2010) metapopulations.