This paper introduces a dynamic formulation of divergence-regularized optimal transport with weak targets on the path space. In our formulation, the classical relative entropy penalty is replaced by a general convex divergence, and terminal constraints are imposed in a weak sense. We establish well-posedness and a convex dual formulation, together with a dual existence result and explicit structural characterizations of primal and dual optimizers. Specifically, the optimal path measure admits an explicit density relative to a reference diffusion, generalizing the classical Schr{ö}dinger system. In the case of zero transport cost, which corresponds to a non-entropic dynamic Schr{ö}dinger problem, we further characterize the flow of time marginals of the optimal bridge, recovering known results in the entropic setting and providing new descriptions for non-entropic divergences, including the $χ^2$-divergence