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Many drift-diffusion transport models rely on a coupling with a sub-model of the drift velocity. In this letter we extend Feynman-Kac's theory to provide probabilistic representations of such velocity-coupled models, so far remained out of reach. Hence a single embedded stochastic process is built, enabling such representations in a single branching path-space. To address this, we propose renewed physical insights in terms of propagative pictures to non-linear physics such as Navier-Stokes, Keller-Segel and Poisson-Nernst-Planck equations in confined and complex geometries.