If a student needs to be moved to a different class within their same school, administrators (not teachers) can move the student by following these steps:
Go to the school's user management page.
Select the student(s) to be moved.
Click on the "Change class" selector and choose the new class for the student(s).
Confirm the change on the pop-out.
Once the change is confirmed, the student(s) will be added to their new class with all of their progress carried over.