Title: Enhance Local Consistency for Free: A Multi-Step Inertial Momentum Approach. Abstract: Federated learning (FL), as a collaborative distributed training paradigm with several edge computing devices under the coordination of a centralized server, is plagued by inconsistent local stationary points due to the heterogeneity of the local partial participation clients, which precipitates the local client-drifts problems and sparks off the unstable and slow convergence, especially on the aggravated heterogeneous dataset. To address these issues, we propose a novel federated learning algorithm, named FedMIM, which adopts the multi-step inertial momentum on the edge devices and enhances the local consistency for free during the training to improve the robustness of the heterogeneity. Specifically, we incorporate the weighted global gradient estimations as the inertial correction terms to guide both the local iterates and stochastic gradient estimation, which can reckon the global objective optimization on the edges' heterogeneous dataset naturally and maintain the demanding consistent iteration locally.   Theoretically, we show that FedMIM achieves the $\mathcal{O}\big({1}/{\sqrt{SKT}}\big)$ convergence rate with a linear speedup property with respect to the number of selected clients $S$ and proper local interval $K$ in each communication round under the nonconvex setting. Empirically, we conduct comprehensive experiments on various real-world datasets and demonstrate the efficacy of the proposed FedMIM against several state-of-the-art baselines.