Sequential Bayesian updating has become the de-facto standard technique for the estimation of stochastic parameters in engineering systems, specifically in dynamic settings. The Bayesian approach estimates a posterior distribution, or a sequence thereof, of a model's parameters based on some prior knowledge and measured data, where the latter can come from a variety of sources. Model uncertainty as well as measurement errors can be factored in seamlessly in this framework. Classically, the Kalman filter and its extensions to nonlinear systems, such as the extended KF or ensemble filters, are employed here. However, many of the algorithms use some assumptions to obtain the estimates, mainly linear systems or Gaussian noise, which in some situations can lead to inaccuracies and limitations. In recent years, through advances in the field of transport theory, new techniques to estimate the arising probability functions have emerged. By formulating the problem in a variational setting, so-called transport maps can be constructed that couple the posterior to a simple distribution, e.g. standard-normal. The map is found via optimization by minimizing a stochastic distance between the approximation and the target. In the present study, we employ a coupling filter that is able to draw samples from intermediate posterior densities in the filtering setting by construction of a coupling between the joint density of system states and data and the standard normal density. This coupling filter has the advantage that no assumption on the structure of the intermediate probability functions is made and all calculations are based on polynomial evaluations which opens up the possibility for real-time inference. We will demonstrate here the application on joint state-parameter estimation of time-dependent data obtained from a small-scale shaking table. The data consists of rigid body displacement measurements obtained by stereo digital image correlation.