A large variety of communication systems, including telephone and data networks, can be represented by so-called Whittle networks. The stationary distribution of these networks is insensitive, depending on the service requirements at each node through their mean only. These models are of considerable practical interest as derived engineering rules are robust to the evolution of traffic characteristics. In this paper we relax the usual assumption of static routing and address the issue of dynamic load balancing. Specifically, we identify the class of load balancing policies which preserve insensitivity and characterize optimal strategies in some specific cases. Analytical results are illustrated numerically on a number of toy network examples.