Starting from full-dimensional models of solute transport, we derive and analyze multi-dimensional models of time-dependent convection, diffusion, and exchange in and around pulsating vascular and perivascular networks. These models are widely applicable for modelling transport in vascularized tissue and brain perivascular spaces. We show the existence and uniqueness of solutions to both the full- and the multi-dimensional equations, and we quantify the associated modelling error with a-priori estimates in evolving Bochner spaces. Further, we propose discontinuous Galerkin (dG) approximations to the 3D-1D coupled systems and prove convergence to weak solutions. A new dG formulation for networks embedded in 3D domains is also presented. Numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our approach.