Slope failure induced by sublevel caving mining is a progressive process, resulting in the large deformation and displacement of rock masses in the slope. Numerical methods are widely used to investigate the above phenomenon. However, conventional numerical methods have difficulties when simulating the process of progressive slope failure. For example, the discrete element method (DEM) for block systems is computationally expensive and possibly fails for large-scale and complex slope models, while the finite difference method (FDM) has a mesh distortion problem when simulating progressive slope failure. To address the above problems, this paper presents a finite difference modeling method using the adaptive local remeshing technique (LREM) to investigate the progressive slope failure induced by sublevel caving mining. In the proposed LREM, (1) the zone of the distorted mesh is adaptively identified, and the landslide body is removed; (2) the updated mesh is regenerated by the local remeshing, and the physical field variables of the original computational model are transferred to the regenerated computational model. The novelty of the proposed method is that (1) compared with the DEM for block systems, the proposed LREM is capable of modeling the progressive slope failure in large-scale rock slopes; (2) the proposed method is able to address the problem of mesh distortion in conventional FDM modeling; and (3) compared with the errors induced by the frequent updating of the mesh of the entire model, the adaptive local remeshing technique effectively reduces calculation errors. To evaluate the effectiveness of the proposed LREM, it is first used to investigate the failure of a simplified slope induced by sublevel caving mining. Moreover, the proposed LREM is applied in a real case, i.e., to investigate the progressive slope failure induced by sublevel caving mining in Yanqianshan Iron Mine.