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AI weather models and weather-based foundation models have demonstrated impressive skills in short- to medium-range forecasts. While most weather models become unstable on longer time scales, a wide variety of AI climate emulators have been proposed, raising questions about the fundamental differences between these approaches.In this work, we compare state-of-the-art models when producing rollouts on annual time scales. We quantify and characterize different types of instability: smoothing, visual artifacts, drift, and loss of seasonality. This analysis highlights the previously unreported stability of the Aurora foundation model and the Earth System Foundation Model (ESFM) for rollouts longer than 35 years.To encompass more diverse representations of possible states of the Earth, ESFM is pretrained on a variety of CMIP6 datasets from the historical period, in addition to the ERA5 reanalysis commonly used in AI models. ESFM also includes climate forcings for physically driven long rollouts. We demonstrate the benefits of CMIP6 pretraining when finetuning on new CMIP6 datasets, including datasets with higher resolution, unseen physical processes, and climate change scenarios.Overall, this work opens perspectives to adapt large-scale pretrained foundation models to the specific challenges of climate projections.