How do I move chapters from one part to another?

From version 1.9.24, press Reorder above the chapter list on the book's page. Parts appear in the list as rows of their own, with their chapters under them. Drag a chapter (or use its arrows) above or below a part's row to move it into or out of that part, or drag a part's row to change where the part starts. Press "Apply new order". A part left with no chapter under it shows "No chapter under it: this part is removed when you apply". Chapter numbers, versions and translations follow the new order, as always.