We propose an approach to using run-time monitoring for the re-engineering of distributed systems. Our re-engineering method MBRE consists of five steps that provide guidance during a re-engineering. By utilizing run-time monitors that are parametric in a policy, we obtain the flexibility needed for a sustainable integration of monitors. We illustrate this and other features of MBRE at three re-engineering case studies in a hypothetical hospital scenario. A key novelty of our approach is that it supports cooperation between monitors also across technological boundaries. This is of high relevance because, for instance, system components in the IoT often run on different platforms and are implemented in different programming languages. Surprisingly, such scenarios have been outside the focus of the run-time-monitoring community so far.