In this paper, we face the challenging issue of defining and implementing an effective law for load balancing in Content Delivery Networks (CDNs). We base our proposal on a formal study of a CDN system, carried out through the exploitation of a fluid flow model characterization of the network of servers. Starting from such characterization, we derive and prove a lemma about the network queues equilibrium. This result is then leveraged in order to devise a novel distributed and time-continuous algorithm for load balancing, which is also reformulated in a time-discrete version. The discrete formulation of the proposed balancing law is eventually discussed in terms of its actual implementation in a real-world scenario. Finally, the overall approach is validated by means of simulations.