Yes, they can! Seesaw supports a unified student journal across classes, grade levels or school years with our Seesaw for Schools service.
Unfortunately, it's not possible to move students to another teacher or class with their journal intact using the free version of Seesaw. Moving students from class to class (or year to year) is only possible with our paid school wide service, Seesaw for Schools.
For teachers using the free version of Seesaw, we recommend that the new teacher add the students as a new student in the new class or new year and start fresh.