Previous studies of adaptive evolution in sink habitats (in which isolated populations of a species cannot persist deterministically) have highlighted the importance of demographic constraints in slowing such evolution, and of immigration in facilitating adaptation. These studies have relied upon either single-locus models or deterministic quantitative genetic formulations. We use individual-based simulations to examine adaptive evolution in a 'black-hole' sink environment where fitness is governed by a polygenic character. The simulations track both the number of individuals and their multi-locus genotypes, and incorporate, in a natural manner, both demographic and genetic stochastic processes. In agreement with previous studies, our findings reveal the central parts played by demographic constraints and immigration in adaptation within a sink (adaptation is more difficult in environments with low absolute fitness, and higher immigration can accelerate adaptation). A novel finding is that there is a 'punctuational' pattern in adaptive evolution in sink environments. Populations typically stay maladapted for a long time, and then rapidly shift into a relatively adapted state, in which persistence no longer depends upon recurrent immigration.