Son and successor
According to the most widely accepted tradition, Kunimitsu was the son of Rai Kunitoshi and his successor at the head of the school; so say the Japanese Wikipedia and the Nihon Kokugo Daijiten (日本国語大辞典, Nihon Kokugo Daijiten). Other traditions make him Kunitoshi’s younger brother, grandson or mere pupil, and Sesko dismisses them as minority views. The traditional genealogy gives as Kunitoshi’s sons, in this order, Ryōkai (了戒, Ryōkai), Kunimitsu and Kunizane. Rai Kunitsugu (来国次, Rai Kunitsugu) appears as Kunitoshi’s son-in-law or as Kunimitsu’s elder cousin, and Sesko supposes that they trained together.
The dates
His signed and dated blades run from Karyaku 1 (1326) to Kan'ō 2 (1351). The Kōsei Kotō Meikan cites one of Shōwa 2 (1313), but Sesko believes that at the time he worked mainly helping his father, and there are works from that period that he signed in his father’s name (代銘, daimei). The only dated tachi known is the one of February 1327 (Karyaku 2), a National Treasure in the Tokyo National Museum. Sesko deduces that he took over the leadership shortly after 1321 (Gen'ō 3), Kunitoshi’s last date, when the latter would have been 82: his masterpieces from 1326 to 1331 arrive without an apprenticeship phase.
The Japanese Wikipedia dates the first blade to Kagen 2, which would be 1304, and gives it as 1327: it is an erratum on the page, because 1327 is Karyaku 2. The Digital Daijisen dictionary places him at the beginning of the Muromachi; the Nihon Kokugo Daijiten, at the end of the Kamakura, like Sesko and Wikipedia.
A second generation?
Genealogies, documents and some blades suggest a Kunimitsu II, but it is not known when the handover would have taken place. The Kotō Meizukushi Taizen places the second around the Kōei era (1342-1345) and Satō Kanzan follows it; Tanobe believes that the differences in the works of Jōwa (1345-1350) and Kan'ō (1350-1352) may be due to the age of a single master. The Kotō Meizukushi Taizen has the first born in 1264 and dying in Shōkyō 4 (1335), at the age of 72; Sesko points out that the Shōkyō era lasted only two years, so it looks like a confusion of eras.
A school that came from Korea
The Meizukushi of the Kanchi-in version (観智院本, Kanchi-in-bon), copied in the Muromachi, says that the ancestors of the lineage came from Korea, and that this is why they were called Rai (来, “to come”). The Kyūshū National Museum observes that, whether or not it is true, it is striking that the school proclaimed a foreign origin. The founder would be Kuniyoshi (国吉, Kuniyoshi), but there is no certain work of his, and the effective founder is Kuniyuki (国行, Kuniyuki).