Coupled climate models are prone to ‘drift’ (long-term unforced trends in state variables) due to incomplete spin-up and non-closure of the global mass and energy budgets. Here we assess model drift and the associated conservation of energy, mass and salt in CMIP6 and CMIP5 models. For most models, drift in globally-integrated ocean mass and heat content represents a small but non-negligible fraction of recent historical trends, while drift in atmospheric water vapor is negligible. Model drift tends to be much larger in time-integrated ocean heat and freshwater flux, net top-of-the-atmosphere radiation (netTOA) and moisture flux into the atmosphere (evaporation minus precipitation), indicating a substantial leakage of mass and energy in the simulated climate system. Most models are able to achieve approximate energy budget closure after drift is removed, but ocean mass budget closure eludes a number of models even after de-drifting and none achieve closure of the atmospheric moisture budget. The magnitude of the drift in the CMIP6 ensemble represents an improvement over CMIP5 in some cases (salinity and time-integrated netTOA) but is worse (time-integrated ocean freshwater and atmospheric moisture fluxes) or little changed (ocean heat content, ocean mass and time-integrated ocean heat flux) for others, while closure of the ocean mass and energy budgets after drift removal has improved.