Not quite; there's a mechanism for this, though the details vary by department/funding source/phase of the moon. Typically one can take at least part of earned grants elsewhere, though sometimes the originating school can still take overhead before transferring the rest (leading to double-overhead, when the new school takes a percentage of the remainder).

(The funding system doesn't always make sense, but it REALLY wouldn't make sense to give a grant, especially something like a CAREER, which is given to a single PI based on the work that PI proposes to do, to a department where the PI who proposed to do the work no longer actually works...)