We have recently developed a model for the high-current breakdown of the integer quantum Hall effect, as measured in contactless experiments using a highly-sensitive torsion balance magnetometer. The model predicted that, for low-mobility samples, the critical current for breakdown should decrease linearly with temperature. This prediction was verified experimentally with the addition of a low-temperature (≲ 300 mK) saturation of the critical current. This saturation is consistent with quasi-elastic inter-Landau-level scattering when the maximum electric field in the sample reaches a large enough value. In this paper we extend this model to nearly integer filling factors to show how the model may account for the shape of the magnetisation signal.