Résumé : New technologies and the deployment of mobile and nomadic services are driving the emergence of complex communications networks, that have a highly dynamic behavior. This naturally engenders new route-discovery problems under changing conditions over these networks. Unfortunately, the temporal variations in the network topology are hard to be effectively captured in a classical graph model. In this paper, we use and extend a recently proposed graph theoretic model -- the evolving graphs --, which helps capture the evolving characteristic of such networks, in order to propose and formally analyze least cost journeys (the analog of paths in usual graphs) in a class of dynamic networks. Cost measures investigated here are hop count (shortest journeys), arrival date (foremost journeys), and time span (fastest journeys).