We present a generic branch-and-cut framework for solving routing problems with multiple depots and asymmetric cost structures, which determines a set of cost-minimizing (capacitated) vehicle tours that fulfill a set of customer demands. The backbone of the framework is a series of valid inequalities with corresponding separation algorithms that exploit the asymmetric cost structure in directed graphs. We derive three new classes of so-called D_k inequalities that eliminate subtours, enforce tours to be linked to a single depot, and impose bounds on the number of customers in a tour. In addition, other well-known valid inequalities for solving vehicle routing problems are generalized and adapted to be valid for routing problems with multiple depots and asymmetric cost structures. The framework is tested on four specific problem variants, for which we develop a new set of large-scale benchmark instances. The results show that the new inequalities can reduce root-node optimality gaps by up to 67.2% compared to existing approaches. Moreover, the complete framework is very effective and can solve instances of considerably larger size than reported in the literature. For instance, it solves asymmetric multi-depot traveling-salesman-problem instances containing up to 400 customers and 50 depots, whereas to date only solutions of instances up to 300 customer nodes and 60 depots have been reported.