In this work we present a branch-and-bound (B&B) framework for the asymmetric prize-collecting Steiner tree problem (APCSTP). Several well-known network design problems can be transformed to the APCSTP, including the Steiner tree problem (STP), prize-collecting Steiner tree problem (PCSTP), maximum-weight connected subgraph problem (MWCS) and the node-weighted Steiner tree problem (NWSTP). The main component of our framework is a new dual ascent algorithm for the rooted APCSTP, which generalizes Wong's dual ascent algorithm for the Steiner arborescence problem. The lower bounds and dual information obtained from the algorithm are exploited within powerful bound-based reduction tests and for guiding primal heuristics. The framework is complemented by additional alternative-based reduction tests. All tests are applied in every node of the B&B tree. Extensive computational results on benchmark instances for the PCSTP, MWCS and NWSTP indicate the framework's effectiveness, as most instances from literature are solved to optimality within seconds, including most of the (previously unsolved) largest instances from the recent DIMACS Challenge on Steiner Trees. In many cases the framework even manages to outperform recently proposed state-of-the-art exact and heuristic algorithms. Since the network design problems addressed in this work are frequently used for modeling various real-world applications (e.g., in bioinformatics), the presented B&B framework will also be made publicly available.