According to my sensei, he has two theories on the Korean origin of KuniYuki / Kuniyoshi. Both ideas concur that the Rai group are of Korean ancestry.
The first is that ancestors or possibly Kuniyoshi himself came to Japan and founded the Rai school. They only started using the "Rai" designation after the decline of the Awataguchi school because the Rai group managed to produce swords cheaper and faster. In theory, the Awataguchi group produced swords with nearly little or no shingane, hence the high quality and also high cost. The Rai School made blades with thin kawagane wrapped around shingane. This was cheaper and faster to make, hence their proliferation.
The second guess is that these Korean ancestors came over to Japan a century or more before the Heian period. They settled in Yamato and later sent a representative at the beginning of the Kamakura period to Yamashiro. Once that smiths became established in the mid-Kamakura, the Rai school solidified itself. I've read that Kuniyuki was good friends, possibly a literal next door neighbor of Ayanokoji Suketoshi, hence the similarities in their workmenship. My sensei has gone further in saying there was a possibility they made have traded works when one was busier than the other and vice versa and signed their own mei's to the other's work accordingly.
Please take with a healthy grain of salt. These were just ideas postulated in discussion between myself and my sensei. This is an excellent thread though that started some good discussion.
Ricky