If you are moving schools and your new institution uses Gradescope, you can have another instructor who has access to the course you will be teaching add you to the course on Gradescope. Another option would be to sign up for a Gradescope account under your new institution using your new institutional email address on Gradescope’s home page.
If you are moving schools and your new institution doesn’t use Gradescope, you can always sign up for free as an instructor on Gradescope’s home page and add your new institution’s name in the School field. Signing up for an account with your new institutional email ensures that every institution’s data is kept separate.