Suppose that you are a graduate student and your adviser tells you, "Guess what! We are moving the lab to more awesome university. Let's go!" What do you do? The possibilities may be broken up by how far along you are within the graduate program.
1st year student. Either you (1) find a new adviser at the old U. or (2) you transfer to the new U.
2nd year student. You may still switch advisers at the old U. You could transfer to the new U and obtain a PhD from the new U. However, there is a slim possibility you could move to new U. but get PhD from old U.
3rd year student. Switching advisers at the old U seems difficult now because you are so entrenched in your thesis work. By now, you've passed your candidacy exam/prospectus, so your options are more open. You could transfer to the new U and obtain a PhD from the new U. Now, there is a stronger possibility you could move to new U. but get PhD from old U, if that's what you wanted.
4th year student. You're likely stay in the same research group, but you may be able to finish all your research at old U. and obtain a PhD from old U., while being advised from afar. Or you could move to the new U. and finish up there, obtaining a PhD from either old or new U.
5+ year student. It is likely that you would stay behind at old U. since you don't have much time left and a move would just set you back. The likelihood of transferring to new U. is low because you may not have enough credits "in-state" from new U. In all likelihood, PhD will come from old U.
I've got a 2nd year and 1st year student. The 1st year decided to switch groups, and we are amicably parting. My 2nd year student is trying the most difficult route. He wants to get the PhD from old U, but work at new U. Because he isn't far along, there all these issues with funding, candidacy exam, health insurance, tuition. UGGGGH. Next post will cover that. Just transfer to new U. already!