Self-sustained combustion oscillations observed at limit cycles in a swirl combustor equipped with a ser- ies of steam assisted liquid fuel injectors are analyzed with the flame describing function (FDF) framework. It is first shown that for globally fuel lean injection conditions, the spray flames investigated burn in a non- premixed mode. A non-sooty and a sooty regime are explored. Their frequency response to acoustic forcing is characterized by exploiting the OH* chemiluminescence signal and a velocity signal measured by Laser Doppler velocimetry at the air injection unit outlet. It is found that the FDF of these non-premixed swirl spray flames features a distinct response compared to the FDF of lean premixed swirl flames. The gain and phase lag of these FDF are both a strong function of the perturbation amplitude. At a fixed forcing fre- quency, the FDF phase lag increases with the perturbation level for the non-sooty flame investigated and it decreases for the sooty flame case. The sooty flame has also a much lower cut-off frequency for the FDF gain than the non-sooty flame. It is then shown that these features are essential to reproduce the correct instability bands and oscillation frequencies observed at limit cycles in the experiments. For the sooty flame, combustion is always found stable in agreement with the low cut-off frequency found for the FDF. A linear analysis for the non-sooty flame fails to capture the low frequency instabilities observed in the experiments that have the largest oscillation levels. Only the use of the FDF yields the cor- rect dynamical states observed in the combustor. It is further concluded that the OH* chemiluminescence signal may safely be used to infer the frequency response of these non-premixed swirl spray flames at globally fuel lean injection conditions.