My current game I've levied the military of two city-states on two separate occasions, both times to block my ally America from beating them senseless, since I didn't have any troops to move to surrounding positions. So I just took theirs, sat them around the city for 30 turns until Teddy got bored (and shot many times by the walls) and left.
All in all, it was a cheaper alternative than pumping out units of my own and moving them into position. I couldn't declare a protectorate war anyway, since America is my ally.
The second time I did it, for Mohenjo Daro, he threw envoys at them and stole suzerain from me, and therefore the levied units. That was annoying. But also no longer my problem, since he lost interest in declaring war on a city state he was now in control of. (Then I took it back, but he must have seen a butterfly somewhere else.)