From Mino to Kyoto
The NBTHK thinks that the first Yoshimichi was a son of Kanemichi (兼道), a swordsmith of Mino, and one of four brothers who moved with their father to Yamashiro: the Mishina school. Markus Sesko specifies that Kanemichi took his four sons, Iga no Kami Kinmichi, Rai Kinmichi, Tanba no Kami Yoshimichi and Etchū no Kami Masatoshi, to Kyoto in 1593 (Bunroku 2), and that they settled in the quarter of Nishinotōin (西洞院); Yoshimichi was the third. Kōdansha’s biographical dictionary, on Kotobank, instead places the arrival of Kanemichi and Kinmichi in Kyoto around the Eiroku era (1558-1570).
Haga Yaichi’s dictionary of names gives the first Yoshimichi the title of Tanba no Kami from 1595 (Bunroku 4).
The five swordsmiths of Kyoto
According to the Japanese Wikipedia, the four Mishina brothers and Ōmi no Kami Hisamichi, a pupil of the house, make up the five swordsmiths of Kyoto (京五鍛冶, Kyō go-kaji), who prospered throughout the Edo period alongside the Umetada and Horikawa schools. According to Hawley, as recorded by the Japanese Sword Index, the Yoshimichi of Kyoto were allowed, from the second generation on, to put the sixteen-petalled chrysanthemum in their signature.
The Osaka branch
Hawley distinguishes two main branches, Kyoto and Osaka, and the Yamada cutting list gives separately the first two Yoshimichi of Kyoto (京吉道, Kyō Yoshimichi) and the first two of Osaka (大坂吉道, Ōsaka Yoshimichi). Kotobank relates that Tango no Kami Kanemichi (丹後守兼道, 1603-1672) was a son of the Yoshimichi of Kyoto, was called 直道 (Naomichi) around the Kan'ei era and later moved to Osaka; Markus Sesko gives him as the second son of the Tanba no Kami of Kyoto, who went to Osaka in the Kan'ei era, and his son as the second Kanemichi. Who founded the branch of the Yoshimichi of Osaka is not recorded in the sources consulted.
The generations
Hawley counts eleven generations of Yamashiro swordsmiths who sign Tanba no Kami Yoshimichi, between 1634 and 1853. Haga Yaichi’s dictionary of names has separate entries up to the seventh: the second, title and permission for the chrysanthemum in 1629 (Kan'ei 6); the third, 1662 (Kanbun 2); the fourth shaved his head in the Genroku era and took a religious name; the fifth, 1711 (Shōtoku 1); the sixth, 1752 (Hōreki 2), and the seventh, 1793 (Kansei 5), swordsmith to the lord of Nobeoka, in Hyūga. Those entries repeat numbers and contradict each other on the order: there is no reliable family tree of the generations in the sources consulted.
The Metropolitan Museum of Art dates a Yoshimichi of Kyoto, the maker of the blades of a daishō for a child, between the end of the 17th century and the beginning of the 18th.