Reef-building corals are extremely important for maintenance of marine biodiversity and coastal economy and are currently under severe threat from anthropogenic warming. Warming is predicted to drive preferential survival of warm-adapted genotypes that have migrated to cooler locations and result in an overall decline in genetic diversity due to bleaching-related mortality. To quantify these trends, we analyzed five populations of a common coral Acropora millepora along the latitudinal extent of the Great Barrier Reef (GBR). Population genomic analysis revealed that most populations were demographically distinct and that migration was indeed preferential southward, from lower (warmer) to higher (cooler) latitudes. However, no recent increase in southward migration was detectable, and inferred migration rates remained closely correlated with predictions of a biophysical model of larval dispersal based on ocean currents. There was also no evidence of recent declines in genetic diversity: populations of A. millepora expanded during the period of rapid sea level changes 100-500 thousand years ago and remained relatively stable since then, with the exception of one population suffering from frequent environmental disturbances. A multi-locus adaptation model indicated that standing genetic variation spread across latitudes should be sufficient to fuel continuous adaptation of A. millepora metapopulation to warming over the next 100-200 years. Unexpectedly, we found that naturally low heritability of thermal tolerance in reef-building corals due to contribution from horizontally transmitted algal symbionts would facilitate longer metpopulation persistence. Still, our model also predicted increase in severity of local mortality events induced by thermal anomalies.