One contribution of 18 to a theme issue 'Human influences on evolution, and the ecological and societal consequences'. The consequences of climate change for local biodiversity are little understood in process or mechanism, but these changes are likely to reflect both changing regional species pools and changing competitive interactions. Previous empirical work largely supports the idea that competition will intensify under climate change, promoting competitive exclusions and local extinctions, while theory and conceptual work indicate that relaxed competition may in fact buffer communities from biodiversity losses that are typically witnessed at broader spatial scales. In this review, we apply life history theory to understand the conditions under which these alternative scenarios may play out in the context of a range-shifting biota undergoing rapid evolutionary and environmental change, and at both leading-edge and trailing-edge communities. We conclude that, in general, warming temperatures are likely to reduce life history variation among competitors, intensifying competition in both established and novel communities. However, longer growing seasons, severe environmental stress and increased climatic variability associated with climate change may buffer these communities against intensified competition. The role of life history plasticity and evolution has been previously underappreciated in community ecology, but may hold the key to understanding changing species interactions and local biodiversity under changing climates.This article is part of the themed issue 'Human influences on evolution, and the ecological and societal consequences'.