mass - How do neutrinos oscillate to more massive states?

Neutrino oscillations show that neutrinos change flavor as they propagate, if I understand correctly, and that this is done by allowing different mass and flavor eigenstates states to mix. Do they have definite mass states when they are measured (or at any time)? If so then how is it possible for a neutrino with a lower mass eigenstate to later have the potential to oscillate into a higher mass eigenstate (or whate makes this not possible)? Is mass and energy conserved by a sudden change in the neutrino's energy?