With conversion to Islam, most Turkic nations and societies trace descent strictly through paternal lines. But, among the Shamanist (even Buddhist, Manachaein and Nestorian Christian) Turkic tribes, descent is through maternal lines. During the Chinese Song Dynasty, the ambassador to the Uygur nation was half-Turkic Uygur by his mother. The Uygur king (Khan), Yaylakar always addressed the Chinese ambassador as "my son."
Korean people always clamor for some association with Mongol/Turkic people, but recent DNA studies show that their paternal genetics are closer to the Han Chinese and Vietnamese, while their maternal genetics are closer to Siberians. I tell them this is good enough qualification because Central Asians trace through mothers line!