Actually, the relationship with Mongolia started with Tulku Drakpa Gyaltsen. I read that there was a fallout between some the Mongol patrons with the Great Fifth Dalai Lama because the Dalai Lama seemed to be patronising and engaging in Nyingma lamas and practices. Mongols by tradition are extremely strict Gelugs and were thus offended with the Dalai Lama's dabbling with the Nyingmas. So, more and more Mongol rich and royal patrons began to favour and seek the other High Lama, Tulku Drakpa Gyaltsen, who had a reputation for being a stickler to Gelug teachings. The conservative demeanor of this Lama pleased and attracted even more patrons and students from Mongolia. This connection grew and some of the surviving biographies and autobiographies of Tulku Drakpa Gyaltsen are surviving only because of Mongol lamas (mentioned by Vajratruth) had kept it while in Tibet, all traces were systematically wiped out by the government of the Gaden Podrang.