Dynamic function optimisation is an important research area because many real-world problems are inherently dynamic in nature. Over the years, a wide variety of algorithms have been proposed to solve dynamic optimisation problems, and many of these algorithms have used the Moving Peaks (MP) benchmark to test their own capabilities against other approaches. This paper presents a detailed account of our hybridised Extremal Optimisation (EO) approach that has achieved hitherto unsurpassed results on the three standardised scenarios of the MP problem. Several different components are used in the hybrid EO, and it has been shown that a large proportion of the quality of its outstanding performance is due to the local search component. In this paper, the behaviour of the local search algorithms used is analysed, and the roles of other components are discussed. In the concluding remarks, the generalisation ability of this method and its wider applicability are highlighted.