The Capacitated Location-Routing Problem consists in, given a set of locations and a set of customers, determining in which locations one should install depots with limited capacity, and for each depot, design a number of routes to supply customer demands. We provide a formulation that includes depot variables, edge variables, assignment variables, and an exponential number of route variables, together with some new families of valid inequalities, leading to a branch-cut-and-price algorithm. The main original methodological contribution of the article is the Route Load Knapsack Cuts, a family of non-robust cuts, defined over the route variables, devised to strengthen the depot capacity constraints. We explore the monotonicity and the superadditivity properties of those cuts to adapt the labeling algorithm, used in the pricing, for handling the additional dual variables efficiently. Computational experiments show that several Capacitated Location-Routing previously unsolved instances from the literature can now be solved to optimality. Additional experiments with hard instances of the Vehicle Routing Problem with Capacitated Multiple Depots and with instances of the Vehicle Routing Problem with Time Windows and Shifts indicate that the newly proposed cuts are also effective for those problems.