jats:p The response of swirl non-premixed flames to air flow oscillations is studied using Large-Eddy Simulation (LES) and the Conditional Moment Closure (CMC) combustion model, focusing on the physical mechanisms leading to the heat release rate oscillations observed in a parallel experimental study. Cases relatively close to blow-off and characterized by different amplitude of the flow oscillations are considered. Numerical results are in good agreement with the experiment in terms of both mean flame shape and heat release rate response. Simulations show that the oscillation of the air flow leads to an axial movement and fragmentation of the flame that are more pronounced with increasing amplitude of the forcing. The flame response is characterized by fluctuations of the flame area, time-varying local extinction and lift-off from the fuel injection point. LES-CMC, due to the inherent capability to capture burning state transitions, predicts properly the flame transfer function as a function of the amplitude of the air flow oscillations. This suggests that the response mechanism for this flame is not only due to time-varying flame area, but also local extinction and re-ignition. This study demonstrates that LES-CMC is a useful tool for the analysis of the response of flames of technical interest to large velocity oscillations and for the prediction of the flame transfer function in conditions close to blow-off. </jats:p>