Generally, @Rarebear & @filmfann would be correct. But yes, @Qingu, your hunch that it would depend on what caused the orbital variance is correct as well. Unstable orbits vary rapidly till they reach a point of stability or the orbiting object is consumed by the gravitational pull of the object it orbits. It’s anybody’s guess what happens to satellites of the object if its orbit is highly unstable. Objects in stable orbits very gradually lose orbital speed. The few hydrogen atoms in space give some resistance, and solar winds add to that, making an apparent headwind the object must move through. Slowing of orbital speed tends to cause the object to spiral closer to the body it orbits. Counterbalancing this, the Sun and other stars lose mass as they convert their fuel into energy. This loss of mass tends to cause objects orbiting them to move further away.
As planets with satellites draw ever closer to a very massive object like a star, the orbit/s of their satellite/s become progressively more disturbed by the massive object’s gravity. Moving further away tends to lessen this effect.
Finally, orbiting planets can have their orbits affected by collisions with large objects. Any collision sufficient to produce a major alteration in the orbit of the planet might have a profound influence on satellite orbits as well.
So the answer truly is, it depends.