It is generally believed that Local search (Ls) should be used as a basic tool in multi-objective evolutionary computation for combinatorial optimization. However, not much effort has been made to investigate how to efficiently use Ls in multi-objective evolutionary computation algorithms. In this paper, we study some issues in the use of cooperative scalarizing local search approaches for decomposition-based multi-objective combinatorial optimization. We propose and study multiple move strategies in the Moea/d framework. By extensive experiments on a new set of bi-objective traveling salesman problems with tunable correlated objectives, we analyze these policies with different Moea/d parameters. Our empirical study has shed some insights about the impact of the Ls move strategy on the anytime performance of the algorithm.