Robustness of distributed routing policies is studied for dynamical flow networks, with respect to adversarial disturbances that reduce the link flow capacities. A dynamical flow network is modeled as a system of ordinary differential equations derived from mass conservation laws on a directed acyclic graph with a single origin-destination pair and a constant inflow at the origin. Distributed routing policies regulate the way the incoming flow at a non-destination node gets split among its outgoing links as a function of the local information about the current particle density, while the outflow of a link is modeled to depend on the current particle density through a flow function. A dynamical flow network is called fully transferring if the outflow at the destination node is asymptotically equal to the inflow at the origin node, and partially transferring if the outflow at the destination node is asymptotically bounded away from zero. A class of distributed routing policies that are locally responsive is shown to yield the maximum possible resilience with respect to the two transferring properties, where resilience is measured as the minimum, among all the disturbances that make 
the network loose its transferring property, of the sum of the link-wise magnitude of disturbances. In particular, the maximum resilience of a dynamical flow network starting from an equilibrium condition, in order to remain fully transferring, is shown to equal its minimum node residual capacity. The latter is defined as the minimum, among all the non-destination nodes, of the sum of the difference between the maximum flow capacity and the initial equilibrium flow on all the links outgoing from the node. On the other hand, the maximum resilience of a dynamical flow network starting from an equilibrium condition, in order to remain partially transferring, is shown to be equal to the network's min-cut capacity and hence is independent of the initial equilibrium flow. Finally, a simple convex optimization problem is formulated for the most resilient initial equilibrium flow, and the use of tolls to induce such an initial equilibrium flow in transportation networks is discussed. 
