It works like this. If you attack the target, and die, the target still has the damage.
The problem is, in these missions, the target continues to travel along the level, or that part of the level, even after you die. (it depends on the stage I think) The target waits for you to reach that part before it begins moving along that part. It will keep moving along that part until it reaches the end, then wait at the start of the part after that. This is all even if you die at any point.
But apparently Air Fleet is an exception. In Air Fleet, if the plane doesn't reach the end of the rail at the outside parts, and you die, then the plane will respawn from the start of that part. I decided to do my run by rushing the checkpoint before the first outside part, hitting the nearby shadow crate for a shadow rifle, unloading it, die, and keep doing it over and over from that checkpoint until the plane blows up.
In Iron Jungle, the blimp moves slowly, so I was able to do it a different way, by unloading a shadow rifle, dying, and so on, while the blimp is slowly moving past the checkpoint I am abusing. (but I still had to finish it off at the next part)
I'm almost certain this is allowed. Nobody told me it wasn't, even after a l33t helped me with it, and the competitors with the best times in these missions obviously took advantage of it. It makes sense to me since even a newbie would almost always inadvertantly end up completing the mission on their first time this way. Avoiding it would require a restart every time you lost a life. (restarting resets damage and target position)
However, the exploit that's not allowed is killing the target then killing yourself. If you time it right, your time will come up as 0:00 if you didn't get a checkpoint. (I accidentally did it this way myself, since it's a hassle to keep yourself alive after killing the plane in Air Fleet. I pulled out my memory card a.s.a.p. ^__^) According to the guide in this site, this expoit was used in Air Fleet. I don't know about other stages.
By the way, IMPORTANT!!: You can't rapid fire shadow rifle at the plane in Air Fleet. It's wasting ammo. (and therefore wasting lives) After each round that hits, the plane is invincible for a very brief moment. I wasn't able to get a nice time in this mission until I was told that in chat. (good thing I whined about shadow rifle feeling weak)