When inferring on the magnitude of future heat-related mortality due to climate change, human adaptation to heat should be accounted for. We model long-term changes in minimum mortality temperatures (MMT), a well-established metric denoting the lowest risk of heat-related mortality, as a function of climate change and socio-economic progress across 3820 cities. Depending on the combination of climate trajectories and socio-economic pathways evaluated, by 2100 the risk to human health is expected to decline in 60% to 80% of the cities against contemporary conditions. This is caused by an average global increase in MMTs driven by long-term human acclimatisation to future climatic conditions and economic development of countries. While our adaptation model suggests that negative effects on health from global warming can broadly be kept in check, the trade-offs are highly contingent to the scenario path and location-specific. For high-forcing climate scenarios (e.g. RCP8.5) the maintenance of uninterrupted high economic growth by 2100 is a hard requirement to increase MMTs and level-off the negative health effects from additional scenario-driven heat exposure. Choosing a 2 °C-compatible climate trajectory alleviates the dependence on fast growth, leaving room for a sustainable economy, and leads to higher reductions of mortality risk.