At least for the NERVA, after shutdown, a residual, thrust-generating cooldown flow of propellant was run through the reactor.
From the engine specs
188.8.131.52.5.1.4 Shutdown and Cooldown Shutdown consists of throttling,
throttle hold, temperature retreat and pump tailoff, and is initiated by a
command signal to depart from rated conditions and is completed ...
Is it currently possible to shut down an actual rocket (not test reactor with external cooling) while in space and then restart it, without melting significant parts of the engine? If so, how?
It was certainly intended that NTRs under development were restartable, but more importantly a clean shutdown of the engine is clearly desirable so you get good ...