Abstract Eastward propagation is an essential feature of the Madden–Julian oscillation (MJO). Yet, it remains a challenge to realistically simulate it by global climate system models, and the reasons are not fully understood. This study evaluates the capability of 20 Coupled Model Intercomparison Project phase 6 (CMIP6) models in simulating MJO’s eastward propagation and its intrinsic links with the dynamic–thermodynamic structures and the background mean states, aiming at better understanding the sources of the simulation errors. The metrics to evaluate the MJO internal dynamics consists of six parameters: 1) the east–west asymmetry in the low-level circulation, 2) the boundary layer moisture convergence propagation, 3) the vertical tilt of equivalent potential temperature or moist static energy, the vertical structures of 4) diabatic heating and 5) available potential energy generation, and 6) upper-level diabatic heating and divergence. We also gauge the performance of three MJO-related background mean-state fields, including precipitation, sea surface temperature, and low-level moist static energy. It is argued that these parameters are relevant internal and external factors that could affect MJO eastward propagation. We find that the boundary layer moisture convergence is most tightly coupled with the eastward propagation of MJO and controls the premoistening, destabilization, and the leading low-level diabatic heating and available potential energy generation. The CMIP6 models exhibit significant improvements against CMIP5 models in simulating MJO dynamic–thermodynamic structures and the mean states. The diagnostics in this study could help to identify the possible processes related to CMIP6 models’ shortcomings and shed light on how to improve simulation of MJO eastward propagation in the future.