Cord-forming fungi form extensive networks that continuously adapt to maintain an efficient transport system, and we can photograph their growth, digitize the network structure, and measure the movement of radio-tracers. Mycelial networks are more accessible than the transport networks of other multicellular organisms, but there are many open questions concerning the coordination of growth and transport within fungal networks. As osmotically driven water uptake is often distal from the growing margin, and aqueous fluids are effectively incompressible, we propose that growth induces mass flows across the mycelium, towards the growing regions. We imaged the temporal evolution of networks formed by Phanerochaete velutina, and at each stage calculated the unique set of currents that account for the observed changes in cord volume, while minimizing the work required to overcome viscous drag. Predicted speeds were in reasonable agreement with experimental data, and cords that were predicted to carry large currents were significantly more likely to increase in size than cords with small currents. We have also developed an efficient method for calculating the exact quantity of resource in each part of an arbitrary network, where the resource is lost or delivered out of the network at a given rate, while being subject to advection and diffusion. This method enabled us to model the spatial distribution of resource that emerges as a fungal network grows over time, and we found good empirical agreement between our model and experimental data gathered using radio-labelled tracers. Our results suggest that in well insulated fungal networks, growth-induced mass flow is sufficient to account for long distance transport. We conclude that active transport mechanisms may only be required at the very end of the transport pathway, near the growing tips. We also developed a simple model of glucose delivery through vascular networks, which indicates that increasing the number of blood vessels in a region can actually decrease the total rate of glucose delivery.