If you need to move a student from one class to another (ex: Advanced Algebra to regular Algebra) and need to move the graded assignments as well,  first go to the main, hosted database and to  Scheduling=>Assign Students to Classes by Class and add the student to the new class; or go to Assign Students to Classes by Student and add the classes to the student. 

If the student has graded assignments in an old class, DON'T remove the student from the old class until you have first moved their assignments to their new class. 

After changing the student's schedule, sync the portal, if you want the changes to occur right away (otherwise it will happen that night.)

To move assignments: go to Gradebook=>Academic=>Gradebook=>By Assignment. Look up the teacher and new class. Click the “Move Student” button on the top toolbar. From the drop down, choose the student, the “Move From Teacher” and “Move From Class”. For each assignment on the list, use the drop down to indicate whether the assignment should be merged with an existing assignment, added to the teacher’s gradebook, or ignored. When done, click the “Import Selected Assignments” button at the top right corner. Let the teacher know that some assignments, with just the new student in it, will show up in her gradebook. When calculating grades, assignments from other classes will only affect the student that is in it.

If the teacher simply wants to add a new student to existing assignments, click the “New Student” button at the top toolbar. Select the Student and choose the assignments you want to add the student to. Then grade the student on those assignments.

After you’ve moved the assignments for the student, you can go back to Schedule Classes (by Class or by Student) and remove the student from the old classes.