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 coordinate-wise minimization of the free energy. In so doing, we obtain finite sample error bounds and exponential convergence of the EM algorithm under a natural generalisation of the log-Sobolev inequality. We further show that this framework naturally extends to several variants of EM, offering a unified approach for studying such algorithms.