You can change the class a student belongs to both from the Administrator Dashboard and the Administrator app (logging into the Playground app with the administrator's email and password). To learn how to do this from the Administrator Dashboard, refer to this article.
How to switch a student between classes
On the Administrator app, start on the Dashboard screen. Tap on the "Student Roster" tile to reach the Student Roster screen. Here, you can use the search bar and the filters to find the student whose classroom status you would like to change. When you find the student, tap on their name to reach their student profile.
On the student's profile, tap the right-facing caret to pull up the "Switch Student's Class" pop-up menu. On this menu, tap the name of the class you would like to switch the student to. You'll then be returned to the student's profile with their class updated under the "Class" section.