Typical density-functional theory (DFT) and approximations thereto solve the many-electron ground-state problem by working from a numerically efficient noninteracting Kohn-Sham reference system and benefit from useful minimization conditions that allow iteration (i.e., self-consistency) to the optimal energy and density. This work reveals that ensembles of ground and excited states can also benefit from similar minimization conditions within ensemble DFT (EDFT) and potential functional theory (EPFT) frameworks. It then reveals that individual excited states also have state-specific stationary conditions that can be deduced from the ensemble solution and apply to EDFT and its EPFT counterpart. However, the state-specific stationary condition for the noninteracting Kohn-Sham EPFT is revealed to be more complicated than the ground-state problem, due in part to a contribution from density-driven correlations that are neglected in typical "ΔSCF"approaches. The importance of this term is scrutinized through exact and approximate examples. Some implications for self-consistency in exact theory and approximations are discussed.