Can't tell you much about NIH, but here at DOE the grant is
 to the institution, not the PI, period.  Having said that,
 DOE retains authority and responsibility to assure that its
 research dollars are wisely spent.  That means DOE has
 authority to take any of several possible actions under the
 circumstances you described.  DOE scientists who manage
 health related grants are always willing to listen to the PI
 and get his or her side of a story, to review any
 significant changes occurring on a grant, and to determine
 if the money could be spent better at a different
 insitution.  The DOE, and I would expect other granting
 agencies as well, has authority to pull a grant back from an
 institution when a PI leaves it, and then spend the money on
 other research, or not spend it at all.  If a new and
 different PI is located at the institution where the
 original PI left, and if DOE is satisfied the new PI can do
 the research as well as the original PI, DOE may leave the
 money with the institution when the original PI leaves.  If
 a PI leaves a funded institution and moves to one without
 funds for the research, DOE may pull back a grant from the
 first the institution and move it to the new one, if the
 receiving institution, and the PI agree to that move.

 It all starts with a call or letter either from the PI who
 is leaving and wants to get money at a new institution, or
 with the institution that wants to change PIs.

 Said with appropriate disclaimers.

 James R. Beall, Ph.D., U.S. Dept of Energy
          xxxxxx@oer.doe.gov