The Kamado family became the inheritors of the Breath of the Sun after Tsukiguni Yoriichi encountered Kibutsuji Muzan and let him escape.
At that time, because Tsukiguni Yoriichi let Kibutsuji Muzan and the demon Tamayo escape, and his close kin in the Ghost Hunter Corps killed the leader of the Corps, he faced blame and was expelled from the Ghost Hunters.
With nowhere to go in the Ghost Hunter Corps and having lost his identity as a Ghost Hunter, Tsukiguni Yoriichi, all alone in the world, came across an ancestor of the Kamado family who was under attack by demons.
Yoriichi saved him and they became good friends.
Two years later, Yoriichi visited this friend again, told him his story, and at his friend's request demonstrated the Breath of the Sun.
The Breath of the Sun that Yoriichi demonstrated then was thoroughly memorized by the Kamado family ancestor, who imprinted all of the movements into his heart.
Thus, the Dance of the Fire God was born within the Kamado family.