ABSTRACT The quenching ‘maintenance’ and ‘cooling flow’ problems are important from the Milky Way through massive cluster elliptical galaxies. Previous work has shown that some source of energy beyond that from stars and pure magnetohydrodynamic processes is required, perhaps from active galactic nuclei, but even the qualitative form of this energetic input remains uncertain. Different scenarios include thermal ‘heating’, direct wind or momentum injection, cosmic ray heating or pressure support, or turbulent ‘stirring’ of the intracluster medium (ICM). We investigate these in $10^{12}\!-\!10^{14}\, {\rm M}_{⊙ }$ haloes using high-resolution non-cosmological simulations with the FIRE-2 (Feedback In Realistic Environments) stellar feedback model, including simplified toy energy injection models, where we arbitrarily vary the strength, injection scale, and physical form of the energy. We explore which scenarios can quench without violating observational constraints on energetics or ICM gas. We show that turbulent stirring in the central $∼ 100\,$ kpc, or cosmic ray injection, can both maintain a stable low-star formation rate halo for >Gyr time-scales with modest energy input, by providing a non-thermal pressure that stably lowers the core density and cooling rates. In both cases, associated thermal-heating processes are negligible. Turbulent stirring preserves cool-core features while mixing condensed core gas into the hotter halo and is by far the most energy efficient model. Pure thermal heating or nuclear isotropic momentum injection require vastly larger energy, are less efficient in lower mass haloes, easily overheat cores, and require fine tuning to avoid driving unphysical temperature gradients or gas expulsion from the halo centre.