Would you ever try cleaning around the movement yourself?

I tried it once on a cheaper skeleton. Used a blower bulb and a very soft brush. It went fine until I touched the balance wheel slightly and the watch started running erratically. Ended up at a watchmaker anyway and cost more than if I'd just taken it in first.

Don't open it. Dust you see through the crystal is often on the inside of the glass, not the movement. And if it is on the movement, a blower can puhs it into pivots. Leave it.

What northbend said. Also, most skeleton watches have very delicate hand-finished bridge. Even a brush with stiff bristles can leave micro-scratches that are permanent. I wouldn't risk it on anything I cared about.

I've done it on a Seagull skeleton I bought used. Removed the caseback, used a rocket blower, then a clean soft brush. No problems, but I was very careful and the watch was not expensive. I would not do it on my Christopher Ward.

The real question is why there is debris inside a sealed case. If it was assembled in a dirty environment, that is one thing. But if it is flaking from a gasket or seal, opening it will not fix that.

I would only ever use a blower, neever a brush. And even then, only if I could hold the movement at an angle so debris falls away from the pivots. Honestly though, the risk is not worth it for me.

Watchmaker here. Please do not do this. The most common damage I sse from owners is bent pivots and scratched bridges from DIY cleaning. A proper service includes disassembly and cleaning in a machine. There is no safe shortcut.

I tried a can of compressed air once. Bad idea. It can spray propellant or condensation. The watch fogged up and I had to get it dried out. Do not use canned air.

Even a blower bulb can be risky if it has talc inside from manufacturing. Some of them do. You would want to flush it first. But again, why bother when a watchmaker can do it properly?

The debris might just be on the underside of the crystal. That is a much easier and safer thing to clean. You can remove the movement from the case without touching the movement itself, clean the crystal, and reassemble. Still requires some care but far less risk.

I have done crystal cleaning on a few watches. It is not hard if you have the right tools and take your time. But I would not go near the movement itself. That is a different level of risk.

If the watch is still under warranty, opening it yourself will void it. Check that first.

I sent mine in for a service after seeing debris. The watchmaker said it was just lint from the factory and cleaned it as part of the service. Cost was reasonable. I would not risk a DIY job on a skeleton unless it was a cheap beater.