If you have grants awarded, and you're moving from one school to the next, then you may consider transferring your awards.
If you're just at the start of the funding period for an award, then transferring is not too much of a hassle. Paperwork is initiated by your old U to transfer the grant in question to the new U. The process is slow, so get started early. There is a potential snag where the transfer takes a while, and a student may not have funding during this gap. In that case, ask for emergency funding to cover the gap (and you can always pay it back when the grant is successfully transferred). The budget of the award may be recalculated according to the new indirect cost.
If you are towards the end of the funding period, then transferring may be more hassle than it is worth. One option is to "spend out" the account, but that could look inappropriate. In some cases, the grant is not transferable because of strings attached (or co-PI's). To maintain access to your grant money, you could set up a sub-award from old U to new U. Or, you could stay on as an adjunct at old U., and incur costs from the grant remotely.
If you have a student staying at old U, then you could either leave some grant money at old U. to fund the student until they graduate or you could transfer the grant and sub-award back to old U to fund that student. A general draw-back with sub-awards is that indirect costs are charged twice; once by new U, and once by old U.
If your student is transferring to new U., then it makes the most sense to transfer the grant to new U.
I found that the best way to explore options was to communicate with grants and contracts representatives at both universities via teleconference. It's a time consuming process. After you've thought about what you want to do, call your program manager and discuss your options. Is your grant transferable? Can you get that in writing? Can you add on a sub-award? Faculty moves are not uncommon, so program managers will know how to handle it.
The whole process is overwhelming at times, but doable. And as for me... my grant is not transferable, and I'm still trying to get a sub-award granted. Sigh.