The only known member was Kozuki Toki, who went by Amatsuki Toki until she married Kozuki Oden over 28 years ago. By marriage, Oden might have been considered part of the family, and similarly by lineage, their children Kozuki Momonosuke and Hiyori may also be considered members.
At least around 800 years ago, a faction of the Amatsuki Family lived outside of Wano Country. Around 800 years ago, a young woman named Amatsuki Toki tried to reach Wano Country and used her ability to move forward in time.
In Wano, the Amatsuki Family were powerful supporters of the ruling family, the Kozuki Family, earning the family heads to be honored with the title of daimyo of a yet unconfirmed region of the country.
At a certain point, Orochi's grandfather, a member of the Kurozumi Family, tried to seize the shogunate and assassinated the leader of the Amatsuki Family as part of his scheme.
Over the course of two years, Toki fell for Oden, and the two became a loving couple. 28 years ago, Toki gave birth to their son, Kozuki Momonosuke, and two years after that, their daughter Kozuki Hiyori was born.
The family stayed with Oden when he joined the Roger Pirates. One day, Toki became too ill to continue their journey and had to stay in Wano with their children, forcing Oden to finish his quest alone.
25 years ago, when Oden returned, he was rejoiced to be reunited with his family and learned that Toki worked tirelessly to not make the citizens of Kuri resent Oden for his absence. He was also informed about Orochi's actions, and in a fit of rage for attacking his family, Oden headed toward the Flower Capital to retaliate.
For the next five years, Toki stayed by Oden's side when the latter humiliated himself on a weekly basis as part of an undisclosed agreement with Orochi.
20 Years ago, when Oden was killed, the Beasts Pirates attacked Oden Castle, with Toki and their children still inside. Toki used her time abilities one more time to send her son and several of their retainers to the future while Kawamatsu escaped with her daughter. Toki stayed behind to send a message to their allies about the time when the Kozuki heir would return.
At the same time, Orochi offered the daimyos to follow him, but they refused and rebelled only to be defeated by Kaido. It is unknown whether anyone from the Amatsuki Family is still alive currently.
Amatsuki (天月,Amatsuki? literally meaning "heaven moon") fits most of the Wano Country families theme of having the kanji for "moon" (月,"moon"?) in its name.
It is one of the only two families related to Wano Country that is known to exist during the Void Century, the other being the Kozuki Family.
It is one of the only two families related to Wano Country that is known to have members live outside of Wano, the other being the Shimotsuki Family.