I have heard the analogy that the spread of karate can be likened to a bouquets of flowers. When they are harvested the flowers are beautiful, but over time they wilt and fade. They lack the roots and soil to continue their growth and reproduction, so over time they die. Karate was transmitted to the world from Okinawa, but in most cases was done so without maintaining the roots. Over time this leads to a hollow version of karate that eventually withers and dies.
Most Okinawan karate teachers have spent decades with their own teachers before starting a new dojo. Even when they have their own dojo they are still connected to their teacher either formally or informally, and in most cases they will still receive guidance until their teacher passes on. All this is done in a common linguistic and cultural matrix. This is the way karate was propagated for hundreds of years before the Japanese, and later Westerners, became interested in karate. After these visitors developed an interest in karate, they stayed on Okinawa for relatively short periods of time, and learned in an environment of an unfamiliar language and culture before going back to their homes and spreading karate.
This is true of all the American servicemen that initially brought karate to the USA. It is also true of Yamaguchi Gogen who founded Goju kai on the mainland of Japan. As for Shotokan while Funakoshi was Okinawan, he was not a top Shorin practitioner of his day, and was chosen to represent karate at a mainland Japan demonstration because he could speak Japanese. I hear someone mentioning Mabuni, again he was never said to be the top practitioner in either Naha-te or Shuri-te. I am fairly certain that these men were very good karate-ka and tried in earnest to transmit their knowledge but like any game of Chinese Whispers the message can and often does become distorted. Purposes change and so then does technique. All of these problems require a correction and that can be returning to the source material.
Many serious budoka seek out the source of their arts. Some kendo players end up in live sword arts, some of those who practice aikido go back aikijutsu, and so on. The idea of looking to the roots of one's art can be seen as part of the path to a fuller understanding of what they do. For the karateka, that path leads to Okinawa. Those who come find it well worth their time to investigate the origins of their style.
To those interested in such an endeavour the question becomes, "What style is the progenitor of my style?" Well, for example, with Goju kai the connection to Goju ryu is clear. Shotokan is offshoot of Shorin.Taekwondo is a mix of Shotokan and local Korean arts, so it traces part of its lineage back again to Shorin. Shitoryu has its roots in both Goju and Shorin. One could do a similar search for the roots of Okinawan kobudo were they so inclined.
In my time here I have met both Japanese and Westerner karateka that have come back to train with their Okinawan source, and I have yet to meet anyone that has left the island disappointed. If you are interested in finding a teacher you could look here, here, or just use this.