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This study is a companion paper to Baillié et al. in which we showed that a past episode of inward migration of Mimas could have created the Cassini Division. We here investigate the possible causes of this inward migration. We suggest two scenarios: one based on a past intense heating of Mimas, and another one on a past intense heating of Enceladus, which would have itself driven an inward migration of Mimas due to a mean-motion resonance. These two scenarios are challenged with numerical modelling of the orbital motion of the satellites, and energy budget, which are confronted to our present knowledge of the interior of Mimas and Enceladus. We show that a past hot Mimas requires an eccentricity of 0.22, while a past hot Enceladus would have needed an eccentricity of 0.25. While the scenario of a past hot Mimas preserves the stability of the mid-sized satellites of Saturn, it threatens Janus and Epimetheus and is inconsistent with the observations of impact basins at the surface of Mimas. However, a past hot Enceladus which would have almost fully melted could be consistent with its differentiated interior, but would probably not have preserved the stability of Tethys, given the high eccentricity required. Both of these scenarios would have challenged the stability of Aegaeon, Methone, Pallene, and Anthe, and implied that the Cassini Division would be younger than 10 Myr. [ABSTRACT FROM AUTHOR]