NIH grants - moving to another institution as PI
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.