How to cope when PhD advisor quits midway
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I went through this experience, and I have friends who've gone through this experience, and it's never fun. You ask a lot of questions; I'll try to answer as many as I can.

  1. What should be the student's considerations when the professor offers him a choice to move along with him?

    Consider the following:

    1. How far along are you in your work? If you haven't proposed yet, it's probably easier just to find a new professor and start anew. If you haven't done any serious research yet (1-2 yrs), definitely find a new advisor and start fresh.
    2. Did you have a good relationship with this person? Do you want to continue working with them?
    3. Often, your credits will not follow you. (I don't have a source for this statement, other than I've been told by numerous people that graduate credits rarely transfer between institutions.) Make sure they will transfer, or that you will be given some sort of pass, before transferring.

    Other stuff (rankings, location, collaborators) should obviously be taken into account as well. In my experience, most students do not move along with their advisor.

  2. Do universities offer coursework waiver and faster-than-usual graduation for students accompanying new professors?

    Almost certainly not.

    Can papers published in the older university be considered a part of the thesis that will be written for the newer department?

    Probably, talk with the university before transferring.

  3. What should a student do if such an option is not available to the professor and also there are not other professors in the department who could/are willing to guide the student? Will the university offer a compensation for him?

    That's pretty unusual. This happens all the time; people are familiar with the situation. In many cases, the department will be willing to help you find someone new. You should view the ordeal as identical to when you chose your initial advisor; you'll probably do (shortened) interviews with a few profs, talk to lab members, look into research, etc. The difference is that, by now, you should be familiar with those people who do research similar to what you've been doing, so your search will be easier; you'll know them, and they should know you, even if only because you've taken a class with them or something.

    You will almost definitely not receive compensation. C'est la vie, my friend... welcome to the real world.