First-generation immigrant populations in industrialized countries frequently have a lower mortality than the host population, a finding that is unexpected and often dismissed as the result of bias. We propose an alternative explanation for a real, albeit temporal, mortality advantage. We base our argument on two premises: First, that there are differences in the progression of the health transition between the immigrants' countries of origin and industrialized host countries; and, second, that there are differences in the speed at which changes in mortality from various causes occur after migration. Mortality from treatable communicable and maternal conditions, still high in many countries of origin, quickly declines to levels close to those of the host country. Mortality from ischaemic heart disease, the most common cause of death in the host countries, takes years or decades to rise to comparable heights. This is because of the time lag between increases in risk factor levels and an increased risk of coronary death. Hence, first-generation immigrants may initially experience a lower mortality than the host population, a point that has so far been under-appreciated in discussions of immigrant mortality. After adopting a western lifestyle immigrants face an increasing risk of ischaemic heart disease. The increase occurs on top of a persisting risk from conditions associated with childhood deprivation, e.g. stomach cancer and stroke--the unfinished agenda of the health transition that immigrants experience.