The "good" thing is that free cities do have loyalty pressure and prevent other free cities to switch to a civilization. Once a few are free, they will not switch back to any civilization. In Online speed setting, the spy's trouble mission take 4 turns, -1 turn with the policy cards, promotion or golden age dedication. A city cannot fight a -35 Loyalty per turn. If we can deal with it, there is other ways to reduce the loyalty. You know the classic moves: kill the governor and the nearby Viktor, put Amani nearby, make sure they are unhappy (+6 go to -6, but hard to control), and if they founded a religion, change it (+3 go to -3), you could manage to make them flip more easily. Or go to war and make them starve*.
But you will probably need the Rock band for that strategy. First you need to train them with +2 level promotions like Glam Rock or Arena Rock (also depend what districts has the target civilization) before going to Indie. You either do a 3 Rock Band performances / 2 Rock Bands + Spy to instantly make the city free (but risky). Or you drain the Loyalty with spies before finishing the job with a Rock band. I think this is the way to go: the difficulty will not be on the city on your border but those are far away. It is improbale you will have all the city within 10 tiles of your empire.
Or, you can also ICS the world to prevent the other civilization to settle cities and reduce the amount of work to be done. The Phoenician can be a good idea for this: +50% Production towards settlers with the Cothon, additional movement for Settlers in the water, you can forward settling on the coast of the same continent without worrying the Loyalty issue. The Capital city deals double loyalty pressure, and since the Phoenician can move their capital around, it is an additionnal tool that can be used for more loyalty pressure!
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@Bangau, the Mapuche are also nice if you go the warmonger route. You could pilllage for -5 Loyalty each time. Tier2 promotion "Depredation" for Light Cavalry and the Malon Raider reduce the pillaging cost to 1 Movement, making this easier to pull off. Plus, pillaging all the luxury ressource, entertainement district and all the farm will put them in starving (-4 Loyalty) and unrest (-6 Loyalty).
Be careful with Eleonor: even if you refuse to take the city, her Great Works are still going to do loyalty pressure and other civilization can grab the city even if they just have Amani around.
*: I wonder if we can "peacefully" make a city starving and facing an unrest. For example, I make an alliance with the civilization while being in Democracy with the Alliance card, put all my Trader to the target city (+6 Food 10 Traders = +60 Food) to make it grows like crazy. Then the alliance stops: the city suddenly have no food while not having the Amenity to sustain a 40+ Population city. You could finish the job with the spies. While free, it isn't sure the city will go back to the former civilization. Due to the large Population, it may prevent it to face too much Loyalty issue.