Blocking over Greenland is known to lead to strong surface impacts, such as ice sheet melting, and a change in its future frequency can have important consequences. However, as previous studies demonstrated, climate models underestimate the blocking frequency for the historical period. Even though some improvements have recently been made, the reasons for the model biases are still unclear. This study investigates whether models with realistic Greenland blocking frequency in winter have a correct representation of its dynamical drivers, most importantly, cyclonic wave breaking (CWB). Because blocking is a rare event and its representation is model-dependent, we use a multi-model large ensemble. We focus on two models that show typical Greenland blocking features, namely a ridge over Greenland and an equatorward-shifted jet over the North Atlantic. ECHAM6.3-LR has the best representation of CWB of the models investigated but only the second best representation of Greenland blocking frequency, which is underestimated by a factor of 2. While MIROC5 has the most realistic Greenland blocking frequency, it also has the largest (negative) CWB frequency bias, suggesting that another mechanism leads to blocking in this model. Composites over Greenland blocking days show that the present and future experiments of each model are very similar to each other in both amplitude and pattern and that there is no significant change in Greenland blocking frequency in the future. However, these projected changes in blocking frequency are highly uncertain as long as the mechanisms leading to blocking formation and maintenance in models remain poorly understood.