We consider the Schr\"odinger bridge problem which, given ensemble measurements of the initial and final configurations of a stochastic dynamical system and some prior knowledge on the dynamics, aims to reconstruct the "most likely" evolution of the system compatible with the data. Most existing literature assume Brownian reference dynamics and are implicitly limited to potential-driven dynamics. We depart from this regime and consider reference processes described by a multivariate Ornstein-Uhlenbeck process with generic drift matrix $\mathbf{A} \in \mathbb{R}^{d \times d}$. When $\mathbf{A}$ is asymmetric, this corresponds to a non-equilibrium system with non-conservative forces at play: this is important for applications to biological systems, which are naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise the solution of both the static and dynamic Schr\"odinger bridge. For general marginals, we propose mvOU-OTFM, a simulation-free algorithm based on flow and score matching for learning the Schr\"odinger bridge. In application to a range of problems based on synthetic and real single cell data, we demonstrate that mvOU-OTFM achieves higher accuracy compared to competing methods, whilst being significantly faster to train.

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