Species respond to climate change with range and abundance dynamics. To better explain and predict them, we need a mechanistic understanding of how the underlying demographic processes are shaped by climatic conditions. Here, we aim to infer demography–climate relationships from distribution and abundance data. For this, we developed spatially explicit, process-based models for eight Swiss breeding bird populations. These jointly consider dispersal, population dynamics and the climate-dependence of three demographic processes—juvenile survival, adult survival and fecundity. The models were calibrated to 267 nationwide abundance time series in a Bayesian framework. The fitted models showed moderate to excellent goodness-of-fit and discriminatory power. The most influential climatic predictors for population performance were the mean breeding-season temperature and the total winter precipitation. Contemporary climate change benefitted the population trends of typical mountain birds leading to lower population losses or even slight increases, whereas lowland birds were adversely affected. Our results emphasize that generic process-based models embedded in a robust statistical framework can improve our predictions of range dynamics and may allow disentangling of the underlying processes. For future research, we advocate a stronger integration of experimental and empirical studies in order to gain more precise insights into the mechanisms by which climate affects populations.This article is part of the theme issue ‘Detecting and attributing the causes of biodiversity change: needs, gaps and solutions’.