We develop a general probabilistic framework for analyzing propagation of chaos in transport ensemble filters (TEFs), a broad class of interacting particle systems that are used to approximate the sequence of state distributions in hidden Markov models given a history of observations. This class of transport-based filtering algorithms includes the widely used ensemble Kalman filter (EnKF). For this class, we identify the limiting mean-field dynamics and establish the first non-asymptotic, high-probability convergence guarantees for TEFs. The ensemble Kalman filter (EnKF) may be viewed as a robust, cheap-to-implement, alternative to the optimal, Bayesian, filter. However, despite its empirical successes, theoretical understanding of its properties, in relation to the optimal filter, is in its infancy. We study the behavior of its mean-field limit in the linear setting. In this setting the EnKF coincides with the Kalman filter itself, for Gaussian initial data. We study the mean-field EnKF with general, non-Gaussian, initial data. Under the assumptions of nondegeneracy of the signal noise and detectability of the signal-observation pair we derive a variational approximation of the Bayes update using a covariance-weighted optimal transport metric, show a strict contraction towards the subspace of Gaussian measures, and deduce a stable form of almost sure accuracy of the mean-field EnKF.