Title: Transport with Support: Data-Conditional Diffusion Bridges. Abstract: The dynamic Schrödinger bridge problem provides an appealing setting for posing optimal transport problems as learning non-linear diffusion processes and enables efficient iterative solvers. Recent works have demonstrated state-of-the-art results (eg, in modelling single-cell embryo RNA sequences or sampling from complex posteriors) but are typically limited to learning bridges with only initial and terminal constraints. Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB). We combine learning diffusion models with Bayesian filtering and optimal control, allowing for constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints. We assess the effectiveness of our method on synthetic and real-world data and show that the ISB generalises well to high-dimensional data, is computationally efficient, and provides accurate estimates of the marginals at intermediate and terminal times. 