Short answers, pulled from the story.
Qachuli founded the Barlas tribe within Khamag Mongol in Northern Mongolia during the early 12th century. He was either the son of Tumanay Setsen or Tumbinai Khan himself and is described as the twin brother of Qabul Khan.
Qarachar Barlas migrated from Mongolia to Transoxiana to establish new settlements while holding the title of Noyan. The Barlas controlled the region of Kish, known today as Shahrisabz in Uzbekistan, and maintained power over settled populations in the valley of Kashka Darya.
Timur founded the Timurid dynasty within the Barlas clan during the 14th century. A portrait painted between the 2nd of May 1405 and the 9th of May 1409 depicts Timur as a member of the Barlas lineage.
Extensive contacts with native Central Asian populations led the Barlas tribe to adopt Islam as their primary religion. They abandoned Tengrism, the faith of their forefathers, in favor of Islamic practices common in the region and gradually adopted the Chagatai language.
Babur descended directly from the Barlas clan through Timur's line of succession and founded the Mughal Empire in Central Asia and South Asia after Timur's death. He established his rule over the Indian subcontinent using his ancestral claim to power, making the Mughal Empire the second major branch of the original Barlas imperial dynasties.