A key decision for every product consolidation node (e.g., warehouse or cross-dock) in a distribution network is the synchronization of inbound and outbound product flows. Among other operational issues, the physical space restrictions as well as the restricted availability of resources (e.g., forklifts) limit the number of vehicles that can be processed at the same time. As a result, the coordination and scheduling of inbound and outbound vehicles becomes a hard problem to solve with significant impact on the overall network performance. This paper introduces a new Vehicle Routing Problem with capacitated Cross-Docking. In particular, the cross-dock capacity represents the maximum number of products that can be handled at the cross-dock for unloading, consolidation, and loading operations, at any point in time. We also consider a First-In First-Out (FIFO) policy for serving the inbound vehicles at the cross-dock. This operational aspect has not been thoroughly examined in the literature. At first, a mathematical formulation is provided. Next, we present a local search metaheuristic algorithm that is tested on existing as well as new benchmark data sets. Various computational experiments are performed to explore the effect of the cross-dock processing capacity on the total transportation costs. From the methodological perspective, the handling of capacity constraints at the cross-dock has several challenges. To deal with the additional complexity, a heuristic mechanism and a Constraint Programming optimizer have been used as alternative approaches for efficiently evaluating the feasibility of tentative solutions. A performance comparison of these approaches is presented and algorithmic insights are provided.