Local search is an integral part of many meta-heuristic strategies that solve single objective optimisation problems. Essentially, the meta-heuristic is responsible for generating a good starting point from which a greedy local search will find the local optimum. Indeed, the best known solutions to many hard problems (such as the travelling salesman problem) have been generated in this hybrid way. However, for multiple objective problems, explicit local search strategies are relatively under studied, compared to other aspects of the search process. In this paper, a generic local search strategy is developed, particularly for problems where it is difficult or impossible to determine the contribution of individual solution components (often referred to as inseparable problems). The meta-heuristic adopted to test this is extremal optimisation, though the local search technique may be used by any meta-heuristic. To supplement the local search strategy a diversification strategy that draws from the external archive is incorporated into the local search strategy. Using benchmark problems, and a real-world airfoil design problem, it is shown that this combination leads to improved solutions.