With fog it is not really advecting and replacing the cold air, it is warm moist air mixing with cold air.
If you have warm aloft and cold air trapped below it, it will be stable, because if a parcel of air is forced to rise, it will be colder than the air it is rising into, so the moment the trigger stops lifting it, it will sink back to the level it started from, because it is more dense. You have an inversion if the temperature increases with height.
If you have cold advected at height, then you have instability. The parcel of air that is forced to rise will be warmer than the cold air, so it will continue to rise after the trigger is removed. The lapse rate is getting greater.