I would call the store you want to transfer to, see if they have space for you- make sure you talk to the manager! And if that manager agrees to it, then speak to your manager. There's a form that you need to fill out, which your manager signs and sends off to the right places. Atleast that's how it works in Canada

if it’s a Starbucks on your college campus there’s a big chance that it’s not a corporate Starbucks, just a licensed Starbucks. meaning that they serve Starbucks in the same way target would have a Starbucks, but without the same benefits n stuff that corporate Starbucks does.

i could be wrong and there may actually be a stand alone Starbucks at your campus, idk. as for asking your manager to transfer, you’d really just talk to them and ask for transfer request forms, reasoning being due to you moving/relocating and needing somewhere close to where you live.