Once a R01 grant is awarded, the institute owns the grant (btw, this is
clearly spell out the the application instruction and most people don't read
it). The institute does not have to relinquish the grant. If the PI wants
to take the grant with them, the PI need to secure a relinquish statement
from the awarded institution.
With tightening of the NIH funding, some schools will rarer screw you then
release the grant, and usually the institute can find someone qualified for
the grant and replace you. It doesn't happen often, but it could happen.
Also, how the PI wants to carry out the R01 research is never controlled
that tightly from the NIH,unlike a industrial research contract.