This paper considers filtering, parameter estimation, and testing for potentially dynamically misspecified state-space models. When dynamics are misspecified, filtered values of state variables often do not satisfy model restrictions, making them hard to interpret, and parameter estimates may fail to characterize the dynamics of filtered variables. To address this, a sequential optimal transportation approach is used to generate a model-consistent sample by mapping observations from a flexible reduced-form to the structural conditional distribution iteratively. Filtered series from the generated sample are model-consistent. Specializing to linear processes, a closed-form Optimal Transport Filtering algorithm is derived. Minimizing the discrepancy between generated and actual observations defines an Optimal Transport Estimator. Its large sample properties are derived. A specification test determines if the model can reproduce the sample path, or if the discrepancy is statistically significant. Empirical applications to trend-cycle decomposition, DSGE models, and affine term structure models illustrate the methodology and the results.