If you submit an NIH grant (e.g. RO1) as PI and then move to another institution prior to study section review, or after an award is made, what are your options under both scenarios? Also, what is the NIH policy regarding such PI changes?
The answer to your question is specific to your funding from the NIH. From the NIH webpage:
Many NIH grant mechanisms allow the principal investigator to bring their grants with them when they change recipient organizations. This is subject to approval by NIH, and there are specific procedures which must be followed in order to make this process run smoothly (and relatively quickly!). If you are a current NIMH recipient, you MUST contact your NIMH Program Official (PO) and Grants Management Specialist (GMS) BEFORE your move, in order to initiate the process of transferring your grant.
This webpage then goes on to describe the steps necessary to transfer the grant.
I'm going to assume that NIH handles things like NSF does. In the latter case, grants can be transferred from one institution to another. Technically, an award is made to an institution and not an individual researcher, and so if your proposal is accepted, then NIH would send the money to your old institution and they can then find someone on campus who can do the work that you proposed. But in practice, if the award is for an individual's research (as opposed to grants made to support, say, the graduate program of a particular department on campus), then universities have an agreement that allows the researcher to take the grant with them to their new university.
The way this is then handled that when (or before) you move, you talk to the people in the grants/sponsored projects office of your new university about transferring your grant. They will walk you through the steps necessary and probably also recommend that you inform your program manager about this. The process is a bit lengthy, in particular if the grant is complex, but not fraught with uncertainty. It just takes work to go through everything everyone will need to know.
At the end of the day, the scenario you describe happens all the time. Don't lose sleep over it -- it can all be sorted out, you just need to talk to the right people.