In this paper we address a problem of the transportation of people with disabilities where customers are served on an almost daily basis and expect some consistency in the service. We introduce an original model for the time-consistency of the service, based on so-called time-classes. We then define a new multi-day Vehicle Routing Problem (VRP) that we call the Time-Consistent VRP. We address the solution of this new problem with a Large Neighborhood Search heuristic. Each iteration of the heuristic requires solving a complex vehicle routing problem with multiple time windows and no waiting time which we tackle with a heuristic branch-and-price method. Computational tests are conducted on benchmark sets and modified real-life instances. Results demonstrate the efficiency of the method and highlight the impact of time-consistency on travel costs.