Understanding how community composition is reshaped by changing climate is important for interpreting and predicting patterns of community assembly through time or across space. Community composition often does not perfectly correspond to expectations from current environmental conditions, leading to community-climate mismatches. Here, we combine data analysis and theory development to explore how species climate response curves affect the community response to climate change. We show that strong mismatches between community and climate can appear in the absence of demographic delays or limited species pools. Communities simulated using species response curves showed temporal changes of similar magnitude to those observed in natural communities of fishes and plankton, suggesting no overall delays in community change despite substantial unexplained variation from community assembly and other processes. Our approach can be considered as a null model that will be important to use when interpreting observed community responses to climate change and variability.