It is a bad idea to join a mission in progress of those in it are about to complete it.
Exactly what you described can happen.
When someone on my team d/c's, the rest of us hunker down and avoid killing the boss, until our team is back in the game again.
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I was in the strike for the entire duration but got dc'd right as he was killed, otherwise normally my friends and I refrain from killing until the member has rejoined, its just garbage how I did the majority of the work and got no rewards