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Dumbfire missiles are designated (DF); all other (guided) missiles have the attribute "retarget". That means if the current target is destroyed, the missile will start tracking nearest enemy.
Missile in flight can be destroyed by weapons. Some fighters with V Crushers may damage themself by launching missile and immediately detonating it with own weapons fire. Skunk missiles may detonate near skunk if launched while under enemy fire.
Interesting thing is that missiles can't damage cargo containers and asteroids.