To be accurate, if the mountain's temp is above absolute zero (certainly true on earth), it's radiating heat even in the daytime, but in the daytime the incoming solar radiation would likely be greater than the outgoing radiation.
Air descends down a mountain slope because air is colder (and therefore denser) at higher altitudes (but only up to about 11km altitude). The air would naturally warm (not cool) as it descends (due to increasing air pressure with lower altitude), even without the mountain. If the mountain is warmer than the descending air, the air will also absorb some heat by conduction from the mountain as it travels down the mountainside. The mountain is also radiating heat, and some of the radiated heat will be absorbed by the katabatic flow, but much will be radiated into the higher atmosphere or even into space. This absorbed heat will certainly contribute to warming the katabatic flow, and when (or if) the flowing air is eventually warmed enough so that it's no longer denser than the ambient air, it will stop flowing downhill. Of course, the momentum of the descending air mass will continue to drive the katabatic flow for some additional distance down the mountain.