Drywall – Why would the mud crack and fall off a ceiling repair

I've seen this happen often at the corners of windows and doors. My first question would be - is there a seam in the plasterboard where the crack is occurring? If not, is the crack just through the skim coat or is the board underneath actually cracked?

If you just plaster over the crack, it will certainly crack again. I would try either taping over the crack with mesh tape and plastering over the tape or if it's just a hairline crack, caulk it and paint it. The caulk should be flexible enough not to crack again.

Firstly, @Michael Karas answer addresses the "sagging"

Cracking: Premixed Joint Compound (JC) hardens by drying out. As such, it tends to shrink. I only use it for final top coats, where the shrinkage is mitigated and application layers are about 1/8 or so.

Setting JC is dry and mixed with water to a similar consistency as Premixed JC, but it hardens chemically (due to plaster of paris). It shrinks much less.

If your patch is dry and firm, you can continue to overcoat it with premix until you obtain a flat surface.

If it is not firm, remove offending parts until you have a firm base. Replace with setting JC, adding fiberglass mesh tape or other backing as needed. Bare JC shouldn't span more than 3/4 inch without a mechanical backing: Refer to Jon Raynors answer here