  We consider a basic model of a dynamical distribution network, modeled as a
directed graph with storage variables corresponding to every vertex and flow
inputs corresponding to every edge, subject to unknown but constant inflows and
outflows. As a preparatory result it is shown how a distributed
proportional-integral controller structure, associating with every edge of the
graph a controller state, will regulate the state variables of the vertices,
irrespective of the unknown constant inflows and outflows, in the sense that
the storage variables converge to the same value (load balancing or consensus).
This will be proved by identifying the closed-loop system as a port-Hamiltonian
system, and modifying the Hamiltonian function into a Lyapunov function,
dependent on the value of the vector of constant inflows and outflows. In the
main part of the paper the same problem will be addressed for the case that the
input flow variables are {\it constrained} to take value in an interval. We
will derive sufficient and necessary conditions for load balancing, which only
depend on the structure of the network in relation with the flow constraints.
