Multi-objective local search (MOLS) algorithms are efficient metaheuristics, which improve a set of solutions by using their neighbourhood to iteratively find better and better solutions. MOLS algorithms are versatile algorithms with many available strategies, first to select the solutions to explore, then to explore them, and finally to update the archive using some of the visited neighbours. In this paper, we propose a new generalisation of MOLS algorithms incorporating new recent ideas and algorithms. To be able to instantiate the many MOLS algorithms of the literature, our generalisation exposes numerous numerical and categorical parameters, raising the possibility of being automatically designed by an automatic algorithm configuration (AAC) mechanism. We investigate the worth of such an automatic design of MOLS algorithms using MO-ParamILS, a multi-objective AAC configurator, on the permutation flowshop scheduling problem, and demonstrate its worth against a traditional manual design.