This paper deals with the stability analysis of BDF Slowest-first multirate time-integration methods applied to the transient analysis of circuit models. From an asymptotic analysis it appears that these methods are indeed stable if the subsystems are stable and weakly coupled.
Verhoeven, A., Maten, ter, E. J. W., Mattheij, R. M. M., & Tasic, B. (2007). Stability analysis of the BDF slowest-first multirate methods. International Journal of Computer Mathematics, 84(6), 895-923. https://doi.org/10.1080/00207160701458641