There are lots of issues with moving one's lab. I've done more than once, and each time I have sworn it will be The Last Time for This Kind of Nonsense.
I was reminded of one of the Big Issues by this post from ragamuffinphd about the problems she's had and is having as the postdoc in a moving lab. Her situation has been ugly, although she has found some support and help from other women at her current institution.
PI's move. Ambitious PI's often move a lot. I've moved more than most, it seems. I started to write a long discussion of the moves, which included to Australia, for a year's sabbatical. Then I realized it would be excruciatingly boring to anyone but me. And besides, its not the point of this post. What I do want to talk about is what happened to my trainees, and how I tried to work on that.
I tried to time each move so students were graduating, postdocs were ready to move on, or projects were coming to a close. Usually it was ok, but sometimes it just didn't work that well. Everyone is the hero of their own story. And I suspect there are trainees who paint me as very evil indeed for when I moved. I did my best, and there were times it just wasn't good enough in the eyes of some people. That is going to happen. But I did learn a few things, things that perhaps might be useful to others.
For PI's moving: do not underestimate the time you will lose due to the move. Even with the best of intentions, a superb team moving with you and great support at the other end, it will impact the flow of data collection, analysis, publication and grant submission. Factor this into your life. But, also, pay attention to the trainees. Talk to them as soon as you can, be honest, and give them as many options as you can. And understand that they may not do what you want. This is called life. Remember this will be true of your children, too. Your goal is not to be liked by your trainees, but to do the best you can by them and their professional growth.
For Trainees: understand that your PI has concerns besides you. Good PI's will do their best for you, but what motivates them, what pushes them may have very little to do with you. They may care about you, but their spouse, their children and most likely their career are all going to come before you in the world. This does not necessarily make them a bad PI or mentor. It might, but, it might not. Just recognize that this is the way of the world. This is akin to realizing that your parents have a life outside of you, and sometimes their priorities will not coincide with yours.
So what to do when your mentor or PI comes and says: "hey, I'm leaving". Firstly, determine if they want you to go with them. Secondly determine if you want to go with them. The answers to these are not obvious. I suspect the most important piece of advice is DO NOT express an opinion in the immediate aftermath of the news. Sit down and think about what it means. Mentally follow the steps through the implications of the changes: what does moving mean to you, to your family, to your expenses? One's first reaction may be emotional, but emotion is not going to serve you well in this situation.
Nextly, do not panic. It may feel like the end of the world, but it's not. As one of my oldest friends is fond of saying: it's an opportunity for growth, but it's time for someone else to have the opportunities. Seriously, figure out what do you want. What is important to you. And those are considerations that are really independent of what your mentor is doing. You will recover. You will find your path. Make a list of the important q's to ask your mentor. As with other interactions, going in to a meeting prepared, with a list, as opposed to going in emotionally charged, can make a huge difference to the ultimate outcome. Things to ask: when will this happen? if I move, does my (salary, position, title, responsibilities) change? Are there any resources ($$) to help me move? Will my lab/office stuff get moved with the larger lab/office stuff? If I don't go with you, what happens? How long can you support me? I am sure there are more, but this is off the top of my head.