Controlling the spread of invasive species, pests and pathogens is often logistically limited to interventions that target specific locations at specific points in time. However, in complex, highly-connected systems, such as marine environments connected by ocean currents, populations spread dynamically in both space and time via transient connectivity links. This results in non-deterministic future distributions of species with local populations emerging dynamically and concurrently over a large area. The challenge, therefore, is to choose intervention locations which will maximize the effectiveness of the control efforts. We propose a novel method to manage dynamic species invasions/outbreaks that identifies the intervention locations most likely to curtail population expansion. Critically, at any point during the development of the invasion or outbreak, the method identifies the local intervention that maximises the long-term benefit across the ecosystem. In so doing, the method adaptively selects the intervention targets under dynamically changing circumstances. We illustrate the effectiveness of the method by demonstrating its potential application to controlling the spread of crown-of-thorns starfish outbreaks across Australia's Great Barrier Reef. We find that application of our dynamic method results in an 18-fold improvement of management outcomes compared to a random targeting of reefs in putative crown-of-thorns starfish control scenarios. Although we focus on applying the method to reducing the spread of species, it can also be reversed and used to help facilitate the spread of species through networks. For example, the method could be used to select those fragments of habitat that are most likely to rebuild a population if sufficiently well protected. This article is protected by copyright. All rights reserved.