Grain boundaries (GBs) significantly affect the mechanical properties of metals and alloys. In this study, we investigated using molecular dynamic simulations the migration behavior of Σ25 (710), Σ5 (310), and Σ37 (750) [001] symmetric tilt GBs in CoCrCuFeNi multi-principal element alloy (MPEA) and Cu samples subjected to shear deformation. In Cu, the migration of the GBs exhibits a coupled migration pattern, consistent with the Cahn model; while in MPEA, the migration pattern varies with GB angle and temperature. Both Σ25 (710) and Σ37 (750) GBs, along with higher temperatures, induce GB roughening and continuous migration in the MPEA samples. Further investigation to the effects of GB angle and temperature was conducted through microstructure evolution tracing and quantitative analysis. A model was developed to describe the temperature-dependent continuous GB migration and average flow stress in MPEA samples with Σ25 (710) or Σ37 (750) GBs. This work can help understand the mechanical behavior of GB in MPEA and provide valuable insights for the development of high-performance materials.