We study the nature and stability of low-frequency waves in collisionless, gyrotropic (P$perp$ ≠ P$parallel$), multi-component plasmas, characterized by arbitrary distribution functions. The ambient medium is assumed to be stratified perpendicular to the magnetic field, with a scale height much larger than the wavelength of our waves, and the whole system is allowed to rotate. We show that the waves are generally non-adiabatic, and we identify the four physical causes of their non-adiabatic behavior. We also derive a global necessary condition for stability, which, quite remarkably, consists of the simple superposition of a criterion against instabilities triggered by thermal pressure anisotropies and a criterion against instabilities triggered by stratification. In the case of a nonrotating bi-Maxwellian plasma, our stability condition turns out to be not only necessary, but also sufficient.