Species' responses to climate change are variable and diverse, yet our understanding of how different responses (e.g. physiological, behavioural, demographic) relate and how they affect the parameters most relevant for conservation (e.g. population persistence) is lacking. Despite this, studies that observe changes in one type of response typically assume that effects on population dynamics will occur, perhaps fallaciously. We use a hierarchical framework to explain and test when impacts of climate on traits (e.g. phenology) affect demographic rates (e.g. reproduction) and in turn population dynamics. Using this conceptual framework, we distinguish four mechanisms that can prevent lower-level responses from impacting population dynamics. Testable hypotheses were identified from the literature that suggest life-history and ecological characteristics which could predict when these mechanisms are likely to be important. A quantitative example on birds illustrates how, even with limited data and without fully-parameterized population models, new insights can be gained; differences among species in the impacts of climate-driven phenological changes on population growth were not explained by the number of broods or density dependence. Our approach helps to predict the types of species in which climate sensitivities of phenotypic traits have strong demographic and population consequences, which is crucial for conservation prioritization of data-deficient species.