Someone brought up Great Physicians in another thread, and it sounds like it'd be a good fit for this sort of idea. I'm not a fan of adding mechanics just for the sake of complexity or micromanagement, though, so it'd have to make the game more engaging and feel fun or interesting to utilize instead of a drag.
I feel like they could rework the development/housing buildings to fit under the same umbrella, namely the Aqueduct and the Sewer. Add in a Hospital and a Treatment Plant and you have the makings of a new district focused on city population. It would allow for offensive and defense population-altering mechanics, such as disease. Cities without sanitary buildings could spawn diseases that would spread through proximity as well as trade, or through random events when you discover a CS or tribal village (successive buildings lowering the chance of a disease spawning). Disease would slow, halt, and then reduce population over XX turns until it is cured or wears off on its own. To prevent them from completely crippling players they could function much like religion--cities with more people exert more disease "pressure" and feel its effects more strongly, like in reality (spread through trade also would work similarly). Spies could poison the water supply as an action in that district to damage population, and such mechanics would be counteracted or lessened in effect by the district buildings themselves (ie: reduce the number of turns for a disease to stop affecting a city). Great Physicians would automatically build a hospital, cure disease in a city, prevent it (via vaccination), heal military units, increase their max health or heal per turn, or in the later eras bioengineer diseases to use against other civs.