We consider the instantaneous equilibrium (ieq) transition of an underdamped Brownian particle under arbitrary time-dependent temperature and potential variations and derive analytic results for the temperature and potential protocols for minimal dissipative work. Explicit results for the time-dependent minimal dissipation protocols and the associated energetics are obtained for the cases of pure temperature variation, pure potential parameter (stiffness) variation, and isentropic ieq processes. The minimal dissipation condition enforces the forward and backward protocols to be time-reversal related, with the same minimal dissipative work. Remarkably, it is shown that the minimal dissipation path is also the isentropic ieq transition. The energetics in the mdieq transitions are analyzed in detail with emphasis on the conditions for maximizing the mean work and power that can be extracted. Furthermore, exact results for the overdamped limit are derived, indicating that there is great freedom to choose the minimal dissipation protocols, thus allowing the realization of mdieq transitions in Brownian colloidal systems with relative ease.