If you update a reviewee's manager after a performance appraisal has been launched, the system will not automatically change the manager. The assumption is that you would want the person's manager at the time of launch to review the person and not the new manager who recently took over the position.
However, if you would indeed like to have the newly assigned manager write the review, you will need to update this manually.
Go to the Appraisals page and press Edit next to the reviewee in question. You will see a blue dialogue box mentioning the new manager. Simply press Update Manager and press the Save button to make the change.