[Andrew]

Go to the top-level Projects list (or the folder that is the parent of the projects that you want to move, if they are already in a folder).

Tap the "Edit" button at the top-right of the screen.

Tap a project that you want to move. The editing controls for that project will be displayed.

Tap the "Move" button at the bottom-right of the screen. This button does not have a label - it is an icon of a folder with an arrow on it, and it is only displayed there in certain situations, like this one.

That will show you a list of folders - tap the desired folder, and the project will be moved there.