I am enjoying using Bread and Circus as an offensive weapon for city flipping!
The only governor that will help on the offensive side of Loyalty is Amani the Diplomat with the Emissary promotion.
Spies can help drop loyalty in a foreign city, and disable a defending governor who is holding out against you.
Mostly it comes down to city size and location, but the above tools can help you out.
Also, Military units are mostly worthless if you are trying to flip a city over to your side They can help clean up the Free City units that spawn. They can also help hold a city once it's flipped to your side, but they provide ZERO loyalty pressure against a foreign city.