There actually some descendents of mongols in many different places far and near Mongolia. I'am not a expert in this field, but many Mongols who settled in the lands as overlords became assimilated into the local populations and only in few places they maintained their language and culture fully.

Foraxample the Hazaras of Afghanistan, they make up around 10-15 % of the country's population. They do not speak the Mongol language, but some traces of Mongolian and Turkic can be found in their dialect of dari (hazaragi). For have been discriminated against for centuries for being descendents of Gjengiz Khan and Shia muslims, it started in time of Babur Shah in the 16th century and recently under the Taliban regime. The word hazar is the persian word for thousand, and the Hazara people is in folk tales seen as desendents of thousand soldiers Gjengiz Khan left behind in this area. Most Hazaras live in Afghanistan, but smaller communities are also found in Pakistan and Iran, they are primarily twelver Shias in contrast to Suni population of Afghanistan. Links to Mongol culture is found in some of their traditions and food culture, when it compared to rest of the Afghans with Pashto and Tajik forefathers.[1]

Some other examples are Kalmyks and Buryats in Russia. The Buryats and other Siberian mongols are close to Mongolian hearthlands, but the Kalmyks are mongols who settled in the Caucasus (Republic of Kalmykia) and stayed there. They not that many only around 170.000 thousand people, but they are for sure descendents of Mongols. And they still speak Mongolian, and have traditions and food culture relatable to Mongolia.[2]

Some would argue that if you look at other ethnic groups seen as Turkic today, may have been of Mongol desendent. I don't have that much insight in this part of the matter, and how the descendents of nomadic steppe people in Russia define themselves. But somebody else could give a good explanation of that.

[1] Bacon, Elizabeth E. "The inquiry into the history of the hazara mongols of afghanistan." Southwestern Journal of Anthropology 7.3 (1951): 230-247.

Dorronsoro, Gilles. "Pakistan and the Taliban: State policy, religious networks and political connections." Pakistan: nationalism without a nation (2002): 161-178.

Mousavi, Sayed Askar. "The Hazaras of Afghanistan: an historical, cultural, economic and political study." Afghan Digital Libraries (1998). Razaiat, Hussain. The Hazara people of Afghanistan: A century of persecution. Aboriginal Catholic Ministry, 2002.

Weires, Michael. "The Language of the Hazara people and of the Mongols of Afghanistan." Afghanistan Journal II 3 (1975): 98-102.

[2] Guchinova, Ė-B. The Kalmyks. Vol. 19. Psychology Press, 2006.

Krueger, John R. "Written Oirat and Kalmyk studies." Mongolian studies(1975): 93-113.

Esenova, Tamara. "Kalmyks in Russia: History, Language, and Culture." 알타이학보 14.단일호 (2004): 139-151.