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FASA Star Trek did something sort of like that. One of the first Earth sublight exploration vessels reached Alpha Centauri, opens a dialogue and gains the Warp Drive Theory. They send the data back at roughly light speed communication, then head home at sublight. Once the message arrives at Earth, the UN sends a second sublight ship to open official diplomatic and trade ties (though trave over a five plus year voyage is a lot). But they also start work on making Warp Drive practical. The first ship then arrives back home about three years later. They develope a viable Warp Driven Explorer and arrive about a month after the UN diplomatic ship arrives several years later. They then take some people back to Earth at Warp Speed for a heroes welcome.
I think Turtledove did that in his World War series, the one where aliens invade Earth in 1942 using sleeper ships. I think it was in the 1960s or something that the Earth sends a sleeper ship to the Alien Empire's homeworld using the same alien tech to start negotiations to a more peaceful coexistance. They arrive some 40+ years later and start talks with the startled aliens. It looks like it might come down to nuclear weapons and a failed mission when an Earth built FTL drive ship arrives to start another kind of negotiations. The "We out tech you now...do you want to talk or die?" negotiations.
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