Exterminations in every captured settlement can leave near all your conquered cities without manpower to replenish your armies.
Which is why you usually do the cities you conquered early on, first. I employ a 'leapfrog' technique to exterminations. I start furthest away from the front lines and work towards them. This provides a steady stream of reinforcements to the front as garrisons can now be thinned out.

There is one advantage to exterminating at the front.......if your losses were light in capturing the city, you don't need to keep much garrison there and the bulk of your army is now free to 'roam about the country.'

Again, I think it's MUCH simpler to periodically exterminate than to mess around with shipping peasants around. And with factions where funds are an issue, the increased farm production puts more money in your coffers. I ALWAYS go for the maximum farm production early on (except in areas that have enhanced grain production) to boost funds, and in trouble areas with low farm output, I'll use buildings that enhance population growth. Later on, when extermination time comes for that particular city, I'll change the temple to something else after the city reaches 24k.

Also I tend to avoid the sewers line of buildings as the public health bonus from these also boosts growth.
Some level of sanitation is required to reduce 'plague-time.' I usually go for high levels here, as well, except for those factions that can build lvl5 plumbing as for some odd reason, the game creators decided that this adds to corruption Go figure.......

Lowering taxes can help, but thats short term strategy. Lower taxes = higher growth rate.
Exactly. I keep every city I own with a 'blue face' for as long as I can get away with it, especially in cities I've just exterminated where taxes get ratcheted up to max and then gradually lowered as revolt nears.

As I said from the beginning, it's too bad one cannot edit the demographics for city growth, as this whole messy business could've been avoided.