Every city can benefit from the skills you have learned in the academy. The only catch is you have to have an academy in every city and your academy in your new city has to be a high enough level. You can see this by opening your academy training window. Each skill will have a score next to it like “Lv. 6/Ct. 6”or whatever. If you see a “Lv. 9/ Ct. 0” it means your skill is at a level 9 but this city doesn’t meet the qualifications yet. It could mean your Academy is too low a level, you don’t have a stable, or forge, or whatever. Look at the cost to learn that skill to find what area you are lacking in. The item you are lacking in will be shown in red text.
As soon as you build the proper structures your city will benefit from the skills you learned in your other academy! For this reason when you build a new city you can quickly speed up production and resources simply by building an academy there. Academy level 1 will speed up food and lumber creation. Level 2 will speed up mining and quarrying.