I placed outpost just outside city, city borders expanded around it. When I built consulate in this outpost, it did not recieve it. So I thought I did not built it there so i did it again (I was spaming outposts and consulates at that time, so I did not remember where I already built it). After some repeatings, I noticed that city was getting inappropriate ammount of growth from consulates. Rush buing builds it temporarily until next load (at which point it disappears from outpost).

Consulates are not actually disappearing, but they are transferred to city itself.

Another obsarvation, it works with High Tower too (so I guess with all outposts improvements)

On load, it works when I rushbuy, next turm save and then load.

Tested it with stables and they were put to city too, but their benefit did not apply. So it looks like that only benefits that can benefit city (range and growth) are exploitable.