The steppe religion did not survive as an institution, because it never was one. It was displaced by religions that had texts, clergy and organisation, and the displacement was gradual, incomplete and largely voluntary.
Islam moved into Turkic Central Asia from the tenth century. The Karakhanid rulers converted around the middle of that century, and Turkic peoples across the region followed over the next several hundred years, with the Golden Horde adopting Islam formally under Ozbeg Khan in the fourteenth. Tibetan Buddhism took the same role among the Mongols, present under the Yuan and decisively established from the late sixteenth century when Altan Khan gave the title Dalai Lama to the third holder of that line and Buddhist institutions spread across Mongolia. Christianity of the Church of the East had a substantial earlier presence among the Kerait, Naiman and Ongud, and several Mongol royal women were Christians.
The steppe complex did not vanish so much as dissolve into these. Sky, earth and mountain rites continued as folk custom under Islamic and Buddhist cover; ovoo cairns are still built and offered at in Mongolia and are handled by Buddhist lamas; Tengri survived as a word for god or sky in several Turkic languages, including in Muslim usage. Ritual specialists persisted, especially in the north, where Buddhism and Islam reached later and less completely.
What did not persist was the political core: a khan ruling by the mandate of heaven, with sky worship as the religion of an imperial elite. That ended with the empires it served. This is the point at which histories most often go astray, treating unbroken folk custom as evidence of a continuing religion of Tengri, when the continuity is in the practices and the discontinuity is in the system that gave them their meaning.