You seem to be a bit confused on the history here ...
Funokoshi did not develop Karate. He helped popularize it by bringing it from Okinawa to Japan and adapting a version of it to Japanese sensibilities.
Karate was developed in Okinawa largely over an extended period when the kingdom, while nominally sovereign, answered to both China and Japan (technically a tributary state to the Ming dynasty while also being a vassal to the Tokugawa shogunate). While Japan did eventually officially annex Okinawa, karate wasn’t specifically developed during that annexation or during the earlier invasion which led to the vassal status.
The change from “China hand” to “empty hand” was not a change in the actual techniques or teaching of the art. It was just a change in the characters used to write the word “karate” in kanji. This was done for political reasons during a time of rising Japanese nationalism.
i would add to Tony's concise post that karate evolves over time and within mainland Japan karate developed its own flavor unique from Okinawa which was of course unique from Chinese fighting styles.
Aubrey if your post is meaning to suggest that karate had its beginnings and evolved from Chinese arts, that would be correct but that is pretty common knowledge. the truth is always more nuanced and complex but yeah as a broad bush yes. Do some reading there is lots to find, Google is your friend.