When a schedule changes the student should automatically update in the gradebook, and grades should transfer.
We are using Aeries for everything and when a student schedule changes (which happens a lot at our school), I had to go in and transfer the students to the other class (I shouldn't have to, when Aeries drops them from my first period and puts them in my 2nd period, it should also do that in grade book), and it should transfer all of the grades that match. I had linked classrooms that students transferred to and I had to re-enter all of the grades prior to the enroll date in the new classroom. What is the point in all of it being connected if I still have to do all of the work.
Within the Aeries Gradebook, there is a feature to be able to transfer scores between linked gradebooks. Here is some information about this feature: https://support.aeries.com/support/solutions/articles/14000067688-gradebook-transfer-scores