I don't know the logistics of NIH GPP, but I have met quite a few PhD students who wanted to transition to MSTP and they had to apply just like anyone else. I would say that unless you get accepted to your school's MSTP program, odds are that you will need to move elsewhere and do research at that new place. Even in PhD programs, I have never heard of anyone doing research somewhere outside of their school (unless the PI moved their lab , but that's a very extreme situation).