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Many species facing climate change have complex life cycles, with individuals in different stages differing in their sensitivity to a changing climate and their contribution to population growth. We use a quantitative genetics model to predict the dynamics of adaptation in a stage-structured population confronted with a steadily changing environment. Our model assumes that different optimal phenotypic values maximize different fitness components, consistent with many empirical observations. In a constant environment, the population evolves toward an equilibrium phenotype, which represents the best compromise given the trade-off between vital rates. In a changing environment, however, the mean phenotype in the population will lag behind this optimal compromise. We show that this lag may result in a shift along the trade-off between vital rates, with negative consequences for some fitness components but, less intuitively, improvements in some others. Complex eco-evolutionary dynamics can emerge in our model due to feedbacks between population demography and adaptation. Because of such feedback loops, selection may favor further shifts in life history in the same direction as those caused by maladaptive lags. These shifts in life history could be wrongly interpreted as adaptations to the new environment, while in reality they only reflect the inability of the population to adapt fast enough.