A student can only belong to one Class per Subject, but can be moved at any time. Their work moves with them and will appear under the new Class in your Markbook.
Method 1: Drag and drop
Go to the Students page and select the student's current Class;
Find the student in the list and hover over their name — the cursor will change to a 4-way arrow;
Drag the student into the target Class in the left column;
A confirmation notification will appear on the right side of the screen.
Method 2: 3-dots menu
Click the 3-dots button to the far right of the student's name;
Select their new Class from the options listed.
Note: To move a student to a different Subject entirely, give them the Class code for that Subject. Their work will not transfer, so save PDFs of their existing work and archive them in the old Subject first.