I have a vague memory of the guidance of the hostile missile actually stop when the UFO is shot down...
Hmm... let me propose simple explanation.
Self-guided (smart) missiles are more expensive than host-guided "dumb" ones. More... it is a real story.
Somewhere in mid-90th airplane's control systems were compared. USAF's one was built with heavy use of ICs and other microelectronics. Nearby nuclear explosion or another source of heavy EM radiation ruins that electronics into useless handful of silicone and copper. Similar Russian systems were built using special vacuum tubes, whose are immune to EM radiation, or almost insensible to it. (Do not tell me about weakness of a glass and so on - I know it)
As to missiles, smart ones have another drawback: lesser warload - dumb ones MAY have larger a bit load of explosives or jet fuel. Let us imagine that aliens use mostly host-guided missiles - their bionics tactical computers are capable to track multiple targets without degrading of performance. But when tactical computer has been powered off because of shot down craft, the dumb missile becomes even more dumber piece of metal, flying according last given directions. And it may also have self-destruction mechanism, which will blow it after the loss of connection to host just to not harm allies.