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Two generations
The Nihon Jinmei Daijiten (日本人名大辞典, Nihon Jinmei Daijiten) places the Yasumitsu in Osafune between the Ōei and Eikyō eras, and distinguishes two generations. The first is, together with Morimitsu, the leading name of Ōei-Bizen; the second is said to have been Morimitsu’s younger brother. No source consulted gives the date of the handover or a documented genealogy of the family.
The earliest dated work located is a wakizashi in the Kyoto National Museum signed 備州長船康光 (Bishū Osafune Yasumitsu) and dated the tenth month of Ōei 16 (1409).
The period
For Markus Sesko, Ōei-Bizen is born in the Ōei era (1394-1428), when slender, elegant blades come back into fashion: its swordsmiths look to the Ichimonji style or to that of Osafune at its best, and refine the wide base (koshi) that the Kozori group (小反, Kozori) had begun. With the Eikyō era (1429-1441) the elements of the temper become smaller, the “Eikyō-Bizen”, a step towards Sue-Bizen. The Japanese Wikipedia relates that at the beginning of the Muromachi the blades called Ōei-Bizen are made, with Morimitsu and Yasumitsu as the outstanding swordsmiths, and calls the Osafune school the largest in the history of the Japanese sword.
An unconfirmed tradition
The description of a photo on Wikimedia Commons attributes to Ōishi Kuranosuke, the leader of the 47 rōnin, a daishō by Kiyomitsu and Yasumitsu of Bizen Osafune, displayed at the Ōishi shrine in Akō (Hyōgo). It has not been possible to confirm it in a museum source or in Wikipedia.