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On the other hand, if you are hosting, then of course your servants will stop doing anything the moment you log off, because the server goes down then.
I've not seen that issue, and I've been sending them out on missions daily for a few weeks now. The timer should be running while you're logged out, i.e. presumably working by reading your system clock and comparing the time vs. the time you initiated the mission. Unless you're closing the game in some wacky way, I can't think of what would cause the mission timer to not run correctly like you're describing.
You might want to post this issue through their suggested bug reporting channels, and ideally try to reproduce what's causing it.