Sending researches over with a conquest, is imo not a very smooth thing to do and very unlogical.
What if you don't take the city? And how would you be supposed to send a research over there and conquer it afterwards? Doesn't make any sense to me.. How would you even be supposed to know what researches you have already sent over there? A special window for that would mean a great coding change as you'd need to make a new attacksort, a new window typefor the city, and a new time frame in which the city needs to get conquerd in order too have the researches implented.. I don't see the devs doing that.
I do like the idea though, a lot, so I suggest you can send researches over to cities you already possess by researches a new research in the academy of the city your sending from called: Send research.
You'd be able to send three researches to the city, and they can only be implented if the city has enough research points and the academy has a high enough level to get them in the first place.. So you wouldn't have to use valuable resources to research them, just spread the love between your cities so to speak..
The way to send researches to your cities should be; pick the city window, click support, and click the "Send research" button. Then you can pick up to three researches to send, but note that the sum of required research points should be available in the city otherwise non will be researched.
In ancient greece the logical thing to do was to use the knowledge you had in other cities you were the leader of, to make the city stronger.. But you wouldn't try and learn a village of farmers and fishers the research of a battleship, they need to be ready to acquire that knowledge before they can actually acquire it.. hence the requirement of having free research points and a high enough academy level.
About the "Send research" research, mentioned before(name can be changed of course), I think it should cost 6 research points and fit in the last tire of the academy researches.
Please give some feedback, I think a pebble's idea is great and we need to get it implented although a pebbles requirements(lvl 1 academy is the only thing you need) are not correct and could be abbused.