In this work, we propose a unifying framework in the space of probability measures for gradient-based and sampling-based moving-horizon estimation methods. We begin with an investigation of the classical notion of strong local observability of nonlinear systems and its relationship to optimization-based state estimation. We then present a general moving-horizon estimation framework for strongly locally observable systems, as an iterative minimization scheme in the space of probability measures. This framework allows for the minimization of the estimation cost with respect to different metrics and divergences. In particular, we consider two variants, which we name \(W_2\)-MHE and KL-MHE, where the minimization scheme uses the 2-Wasserstein distance and the KL-divergence respectively. The \(W_2\)-MHE yields a gradient-based estimator whereas the KL-MHE yields a particle filter, for which we investigate asymptotic stability and robustness properties. Stability results for these moving-horizon estimators are derived in the distributional setting, against the backdrop of the classical notion of strong local observability which, to the best of our knowledge, differentiates it from other previous works. We also present results from numerical simulations to demonstrate the performance of these estimators.