Oh my research was done but I was busy I can send you the book if you want my curiosity has faded Anyway long story short is he is not descendant directly fro Chingiz Khan his hero so to speak he is descendant from what it seems like a Mongolian Family that have been Turkicized Temrs marriage to a Chinggisid princess the daughter of Qazan Khan gave him the right to use the title krgn or imperial son in law which was also adopted by several of his descendants and he attempted to revive the Chinggisid house through the elevation of puppet khans in the Ulus Chaghadai my source is the new Cambridge History of Islam I can send it to you if you like on the Mughals part they see themselves as Timurids but the correct word is Gurkani a simple google book search has concluded me that Baburs diary called Baburnama edited by Thackston in  may contain the Gurkani material