Understanding how species respond to climate change is critical for forecasting the future dynamics and distribution of pests, diseases and biological diversity. Although ecologists have long acknowledged species' direct physiological and demographic responses to climate, more recent work suggests that these direct responses can be overwhelmed by indirect effects mediated via other interacting community members. Theory suggests that some of the most dramatic impacts of community change will probably arise through the assembly of novel species combinations after asynchronous migrations with climate. Empirical tests of this prediction are rare, as existing work focuses on the effects of changing interactions between competitors that co-occur today. To explore how species' responses to climate warming depend on how their competitors migrate to track climate, we transplanted alpine plant species and intact plant communities along a climate gradient in the Swiss Alps. Here we show that when alpine plants were transplanted to warmer climates to simulate a migration failure, their performance was strongly reduced by novel competitors that could migrate upwards from lower elevation; these effects generally exceeded the impact of warming on competition with current competitors. In contrast, when we grew the focal plants under their current climate to simulate climate tracking, a shift in the competitive environment to novel high-elevation competitors had little to no effect. This asymmetry in the importance of changing competitor identity at the leading versus trailing range edges is best explained by the degree of functional similarity between current and novel competitors. We conclude that accounting for novel competitive interactions may be essential to predict species' responses to climate change accurately.