Summary We present a new framework for analysing the expectation-maximization (em) algorithm. Drawing on recent advances in the theory of gradient flows over Euclidean–Wasserstein spaces, we extend techniques from alternating minimization in Euclidean spaces to the em algorithm, via its representation as coordinatewise minimization of the free energy. In so doing, we obtain finite-sample error bounds and exponential convergence of the em algorithm under a natural generalization of the log-Sobolev inequality. We further show that this framework naturally extends to several variants of the em algorithm, offering a unified approach for studying such algorithms.