Human activities are exposing organisms not only to direct threats (e.g. habitat loss) but also to indirect environmental pressures such as climate change, which involves not just directional global warming but also increasing climatic variability. Such changes will impact whole communities of organisms and the possible effects on population dynamics have raised concerns about increased extinction rates. Conservation-minded approaches to extinction risk vary from range shifts predicted by climate envelope models with no population dynamics to population viability analyses that ignore environmental variability altogether. Our modelling study shows that these extremes are modelling responses to a spectrum of environmental sensitivity that organisms may exhibit. We show how the survival curve plays a major role in how environmental variability leads to population fluctuations. While it is often supposed that low-fecundity organisms (those with high parental investment) will be the most vulnerable to climate change, it is those with high fecundity (low parental investment) that are likely to be more sensitive to such changes. We also find that abundance variations in high fecundity populations is driven primarily by fluctuations in the survival of early life stages, the more so if those environmental changes are autocorrelated in time. We show which types of conservation actions are most appropriate for a number of real populations. While the most effective conservation actions for organisms of low fecundity is to avoid killing them, for populations with high fecundity (and low parental investment), our study suggests conservation should focus more on protecting early life stages from hostile environments.