Why do neutrinos propagate in a mass eigenstate? I am aware that flavor $\neq$ mass eigenstate, which is how mixing happens, but whenever someone talks about neutrino oscillations they tend to state without motivation that when neutrinos are actually propagating, they are doing so in a mass eigenstate. Presumably this is glossed over because it is a deep and basic artifact of quantum mechanics that I'm missing, but I'm having trouble coming up with it.
I had some help here where they say 

The mass eigenstates are the free-particle solution to the wave equation[...]

but I suppose "why is that" could be a more basic reformulation of my question! Why can't mass be time-dependent in the wave equation, yielding (I would think) eigenstates that don't have well-defined mass?
 A: It is not true in any sense that neutrinos "have to propagate as mass eigenstates". Quite on the contrary, one of the universal postulates of quantum mechanics – which are valid everywhere in the Universe – is the superposition principle. Whenever the states $|\psi_1\rangle$ and $|\psi_2\rangle$ are allowed, the general linear superposition
$$ a_1|\psi_1\rangle + a_2|\psi_2\rangle $$
is equally allowed as well. The only special feature of the energy (Hamiltonian's) eigenstates is the fact that their time dependence is "simple",
$$|\psi(t)\rangle = \exp(Et/i\hbar) |\psi(0)\rangle $$
This time-dependence boils down to the fact that the Hamiltonian is the operator that generates time translations. Equivalently, it's the operator that appears in Schrödinger's equation or Heisenberg's equations of motion. The reason why energy is connected with the evolution in time is ultimately linked to the deepest definition of the energy we have: via Noether's theorem, energy is the quantity that has to exist and be conserved as a direct implication of the time-independence of the laws of physics.
Note that the energy eigenstates only evolve by changing their overall phase. The phase isn't directly observable which implies that all observable properties of a Hamiltonian eigenstate are actually static in time; they are time-independent. This property makes them very useful as a basis of the Hilbert space: we may immediately solve Schrödinger's equation for this basis i.e. determine how a general state (when written as a combination of energy eigenstates) evolves in time.
These statements are completely general. They're true for neutrinos, too. Neutrino states may also be expanded into a superposition of energy eigenstates which makes the time dependence "easy". In fact, as long as we neglect the interactions, the neutrinos' wave functions evolve in time a a combination of plane waves
$$ \exp(i\vec k \cdot \vec x - i Et) $$
where $E^2-|\vec k|^2=m^2$ is the squared mass. These plane waves are only a useful basis of all the allowed spacetime-dependent wave functions of the neutrinos if $m^2$ is effectively an ordinary number. In general, $m^2$ is another operator that acts on the Hilbert space of states of a neutrino. However, on the subspace of this Hilbert space of eigenstates of $\hat m^2$ with a given eigenvalue $m^2$, the operator $\hat m^2$ may be replaced by the eigenvalue $m^2$. The corresponding plane waves that fully dictate how the neutrino's wave function depends on space and time have a simple form in this case and all allowed wave functions may be written as linear superpositions of these simple plane waves.
A: I keep repeating myself in these answers. Basically Physics does not answer ultimate "why" questions. It really answers how from certain principles and assumptions the mathematical formulation that describes the data will also predict new phenomena. How, not Why.
Why questions on physics become nested as you have seen yourself, when you found an answer to your first question :

The mass eigenstates are the free-particle solution to the wave equation

you go on to ask 

"why is that"

The ultimate answer to why in physics ends up in because  that's the way experiments tell us  that nature is.
@Lubos has answered the basic part, that a free neutrino  can be in a linear combination of the allowed mass states. 

Neutrino oscillation arises from a mixture between the flavor and mass eigenstates of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different superposition of the three neutrino states of definite mass. Neutrinos are created in weak decays and reactions in their flavor eigenstates[nb 1]. As a neutrino propagates through space, the quantum mechanical phases of the three mass states advance at slightly different rates due to the slight differences in the neutrino masses. This results in a changing mixture of mass states as the neutrino travels, but a different mixture of mass states corresponds to a different mixture of flavor states. So a neutrino born as, say, an electron neutrino will be some mixture of electron, mu, and tau neutrino after traveling some distance.

The way we can detect neutrinos is by their interactions, and these are signed by the flavor of the neutrino when it interacts in the detector. An electron neutrino will create an event with an electron, a muon neutrino will manifest a muon in the interaction, and a tau neutrino , a tau. Neutrino mass itself, from missing mass in other interactions of elementary particles was thought to be zero, because it was within the errors of the experiments. It was the observation of oscillations that made it necessary to postulate a mass, small, for the neutrinos.
Neutrinos are not the only ones oscillating. Oscillations were first studied with K0 short and K0 long  and a mass difference was detected for the two states.An article for oscillations of neutral particls  can be found here.
