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Abstract Understanding the demographic drivers of range contractions is important for predicting species' responses to climate change; however, few studies have examined the effects of climate change on survival and recruitment across species' ranges. We show that climate change can drive trailing edge range contractions through the effects on apparent survival, and potentially recruitment, in a migratory songbird. We assessed the demographic drivers of trailing edge range contractions using a long‐term demography dataset for the black‐throated blue warbler (Setophaga caerulescens) collected across elevational climate gradients at the trailing edge and core of the breeding range. We used a Bayesian hierarchical model to estimate the effect of climate change on apparent survival and recruitment and to forecast population viability at study plots through 2040. The trailing edge population at the low‐elevation plot became locally extinct by 2017. The local population at the mid‐elevation plot at the trailing edge gradually declined and is predicted to become extirpated by 2040. Population declines were associated with warming temperatures at the mid‐elevation plot, although results were more equivocal at the low‐elevation plot where we had fewer years of data. Population density was stable or increasing at the range core, although warming temperatures are predicted to cause population declines by 2040 at the low‐elevation plot. This result suggests that even populations within the geographic core of the range are vulnerable to climate change. The demographic drivers of local population declines varied between study plots, but warming temperatures were frequently associated with declining rates of population growth and apparent survival. Declining apparent survival in our study system is likely to be associated with increased adult emigration away from poor‐quality habitats. Our results suggest that demographic responses to warming temperatures are complex and dependent on local conditions and geographic range position, but spatial variation in population declines is consistent with the climate‐mediated range shift hypothesis. Local populations of black‐throated blue warblers near the warm‐edge range boundary at low latitudes and low elevations are likely to be the most vulnerable to climate change, potentially leading to local extirpation and range contractions.