No, they don't. The Kazakhs and the Mongols both take their origin from Altai group, a linguistic (like Indo European language group ) group that is believed to have originated in the Altai region. But one are not descended from the other.
Mongols
The Kazakhs are one of the Turkic peoples. The Turkic peoples are an ethno-linguistic group that is believed to have originated in the area from central Asia to Siberia, including large parts of modern China.
today the Irish trace their ancestry to the Celts.
By the early 20th century the Gobi desert was inhabited by the Mongols, Uyghurs & Kazakhs.
All dogs can trace their ancestry back to the wolf.
China
The majority of people in the United States and Canada trace their ancestry to Europe.
Germany.
Ethnic Russians (Slavs), Ukrainians (Slavs), Belarusians (Slavs), Mongols, Turks, Kazakhs, Koreans, Yakuts and some indegenous people.
Try www.ancestry.com
through their mothers
Muslims trace their ancestry through Abraham's son Ismael. Jews trace their ancestry through Abraham's son Isaac.
I think Kazakhs are nice people.