What would cause the oscillation?

If you're not smack bang in the middle of the galactic plane, you're going to be accelerated towards it by gravity, so you'll tend to bob up and down, like a pendulum.

I'm not sure I buy that the oscillation around the plane can have a period other than the orbital period.

The Radcliffe wave is oscillating through the plane of the galaxy. https://en.wikipedia.org/wiki/Radcliffe_wave

And a recent paper on its oscillation - https://www.sci.news/astronomy/oscillating-radcliffe-wave-12...

The oscillations are not "around" the plane, they are through the plane.

That is; the plane itself has a local gravitational attraction which is orthogonal to the galaxy core gravitational attraction. If a mass is above or below the plane, the "local" gravity will pull towards the plane. since nothing stops it at the mid line of the plane it passes through to the opposite side; rinse and repeat.

This bouncing above and below center line of the disk is more or less independent of our solar system completing galactic orbits.

This is the best explanation IMO.

Fair; the descriptions I've seen seem to focus on the galactic orbit more than the local neighbourhood as the cause of the oscillation; this makes more sense.

I haven't done the math but I would think it's possible due to interactions from other solar systems. Similar to how satellites oscillate around Lagrange points

do you think motion in X is always dependent on motion in Y?

The up down oscillation is completely decoupled from the orbit itself, assuming the orbital potential is uniform. It's like the galaxy wasn't rotating at all. You put something above the disk, and it gets accelerated downwards, but doesn't stop at the disk, it keeps going, until it gets dragged back up. Think of it like a pendulum.

Would we expect the oscillation to get smaller over time, like a pendulum, or is that stretching the analogy too far?

There might be some gas in the way that would slow it down slightly, but I expect that to be a very long time scale compared to other things like interactions with other systems and even the upcoming collision with Andromeda.

Ah that's probably where I learnt it from then!