The objective of this work is to simulate with Large-Eddy Simulation and the Conditional Moment Closure combustion sub-grid model the lean blow-off (LBO) curve of a realistic liquid-fuelled flame. The simulation provides sufficient detail to explore the physics of LBO phenomena relating to: flame structure, intermediate species behaviour, fuel evaporation and fuel starvation. Three kerosene spray flames studied previously experimentally in the Cambridge bluff-body swirl burner are simulated and it is found that LES-CMC reproduces the experimental LBO points within 20% accuracy. The blow-off duration falls in the range 10-30 ms and asymmetric flame structures are observed before blow-off, both comparing well with experiments. CH2O was observed to enter the recirculation zone from downstream during LBO and was present in regions of low to intermediate temperature. The temperature is observed to reduce in the recirculation zone, thus evaporation cannot keep up in the flame region. Reduced temperature is also an obstacle to the pyrolysis of the kerosene vapour, so the flame is starved of fuel, causing the stoichiometric mixture fraction isosurface to shrink down toward the bluff body until it exists only around the injected spray. Fuel starvation is a significant factor in the blow-off of spray flames.