The assumptions about the Lagrangian points being stable (...in the traditional meaning of the word: not moving around; they are unstable in the mathematical sense, except for arguably L4, L5) that I've commonly encountered are:

* a pure 2-body system, with any satellites that sit at these points being of negligible mass.
* the central body is vastly more massive than the orbiting one
* the orbiting one moves in a circular orbit.

I wonder though, do they remain reasonably stable - allow for a satellite to sit there with only minimum of station-keeping, in cases of:

1. the planet's orbit being eccentric, not a perfect circle
2. The two massive bodies being of comparable mass, orbiting a common barycenter (though both in circular orbits). Specifically, how would they morph/move?

bonus: what about a ternary system, where a distant planet orbits something like a contact-binary star? Does the idea of Lagrangian points assume uniformly round celestial bodies?