Why does my Gold just vanish?

-If you're running Universal Suffrage with a governor automating the city they'll rush buildings every so often when you are raking in a huge gold surplus in order to quickly improve those cities. Having a large gold surplus per turn or hefty bank (it seems to be some combination of both) triggers this as long as they still have improvement buildings to construct.

-If you turn your science slider all the way down, or cripple your economy entirely, they will build research/wealth respectively, often to the detriment of relieving the problem instead by actually building something useful like Markets or Courthouses.

-On the subject of relief, I'm not sure if they whip anything but they definitely never do it in order manage happiness, and with citizen automation they'll even avoid growing too big on purpose. I find it more useful to overgrow cities if I am too busy to manage them and I'm still in Slavery; the extra population is free production and the right amount population shaved on a big enough project both eliminates the unhappiness and doesn't waste the hammers, for the least possible.

-They don't read specialist economy strats very well and think you are crippling your city by running too many. This leads to them tending to make odd decisions when picking tiles to work or picking a new building (ex.: low on production in a GP farm? Let's build a quite expensive Forge at 1 hammer per turn! >_>)

-Specialist assignment works the same if you turn citizen automation on; run your EP slider at 0% with good science/gold standing and a lack of courthouses/jails/what-have-you, and every time you get a new citizen to work a fully staffed city or capture an enemy one (where citizen automation is on by default after you take it), they'll run spy specialists, often tons of them. This sucks because they don't usually change their minds later, screwing up GP generation if you don't watch them.

-They build culture in newly founded/captured cities if you have Music, run artists if you have Caste System active, or build Monuments/Theaters in order to get the first border pop.

-Building infrastructure in new/captured cities quickly when you have a lot of tech unlocking buildings at that point. The governor's #1 agenda at default (no focus selected) when tasked to virgin city is to get it online and profitable as soon as possible, so it will build stuff like courthouses, granaries, lighthouses, harbors, etc. quickly but not necessarily other things like markets/libraries as they react to your various incomes as mentioned before. If you are earning plenty of science and gold, they may stop after getting the city some basic things built and just build research.

-They are quite handy for popping borders quickly in captured cities. This is often the most pressing task if you plan to keep a city near enemy territory where their borders get problematic

-They reduce micromanagement load by always keeping your cities working on something. Especially useful in ongoing wars where you just want them to leave you alone so you can focus on the front, and to make sure they resume doing whatever without having to dance around in the queue only to come back and adjust it later.

-Similarly, if you turn on citizen automation with it, they often (not always) pick tiles fairly efficiently, adapt to the task/situation at hand (they work more commerce when your economy is hurting, etc) and they even update their picks as tiles are improved or their projects change. You can adjust the tiles of many cities at a time by multiple selecting and playing around with the various focus options. While not as perfectly efficient as picking the best tiles at the moment yourself, it saves a *ton* of time, especially as games go later and you have lots more to do than fiddle with every city's tiles each turn.

-They suggest options to work on when you may not have a clear task for the city to fulfill at the moment. And for newer players like myself, it helps you understand the usefulness of certain buildings in relation to each other or instead of just building research/wealth all the time. I never figured how important barracks can be until I started coming out of the newbie "peaceful builder" shell and started warring more; consequently, the governors prioritize them in early cities and they help A LOT in keeping you from being distracted by other less useful projects at the moment.