I think the best analogy is birefringence (for 2 flavors). You have 2 orthogonal states of polarization (subbing for flavor), $H$ and $V$, and say electrons only interact with $H$ and muons with $V$, and never the twain shall meet.
Until they start propagating, in a birefringent medium. It has two eigenstates (the fast axis and the orthogonal slow axis, with two different indices of refraction). The index of refraction is sort of a like and effective mass, as it affects the speed of propagation. Moreover, $\Delta n$ plays the role of $\Delta m^2$, the difference in the masses (squared) of the mass eigenstates.
A "non-coincidence of eigenstates" means the crystal is not aligned:
$$ |H\rangle = \cos\theta|F\rangle+\sin\theta|S\rangle$$
$$ |V\rangle = -\sin\theta|F\rangle+\cos\theta|S\rangle$$
It would be a good exercise to fully develop the analogy, but in once $|H\rangle$ starts propagating, there is a phase shift between the $|F\rangle$ and $|S\rangle$ that cause the plane of polarization to oscillate between $H$ and $V$.
Throw in an third flavor, and of course, it gets a bit more complicated, but the idea is the same.