How stable are orbits of light objects around twin large bodies?

Question

I know the two-body problem has a stable solution and the three-body problem does not.

In the case that there are two comparable large bodies (twin planets) in a stable mutual orbit, what happens to a small, light particle's trajectory in that system?

If there were a number of tiny moons around either of these planets, would they remain in stable orbits, or would small petruburances cause all the tiny moons to fall into one or the other large bodies in "short" time?

EDIT: To clarify "stable", I mean that although the trajectory of any tiny moon is likely to be chaotic, it will "probably" avoid colliding with either planet for a "long" time. I realise this is delegating to other imprecise terms but consider that if we gave our own moon a sudden kick which changed its velocity by a few percent we would not expect it to collide with the earth within a decade but rather find a new, different, long-term-stable, approximately elliptical orbit. Could the same be said for a tiny satellite in a twin planet system?

Answer 1

For those moons to remain in orbit around one of those twin planets they most be orbiting within the Hill sphere.

As you can see in the description, for two equal mass planets in a circular orbit with distance $D$ between them, this is $$ R_H = \frac{\small 1}{\small 2} \sqrt[{\Large 3}]{\frac{\small 1}{\small 6}} \ D \approx 0.28 \ D $$ and the article also shows that this is related to the Lagrange points of the twin planets, the orbit of the moons should never come into the neighborhood of those points.