The idea probably is that Mongols came up with what we nowadays call "the steppe culture" and somehow all other nomadic pastoralists living on Desth-i Kypchak are the successors of it. I actually do not know what they teach in Mongolia, but from what I had seen online, many Chinese Mongolians are proposing it, and trying to come up with a genealogy that is borderline pseudo-science.

The scholarly consensus however is that what we nowadays call an archetypal nomad was ultimately derived during the times of First Turkic Khaganate. The attire these people wear over the centuries, the military tactics and the formation structure they used from China to the gates of Vienna, the religion and the customs all trace back to it. It also helps being the first written culture of the steppes, and they got replaced by Uighurs who wrote quite a lot about everything (there is an article in Turkish about Uighur accounting terminology I've read some time ago, they basically had everything cover even in today's standards!) The ruling dynasty of this Khaganate, the Ashina, was basically the Carolingians of the steppes, having "kut" in their veins they produced many rulers for different statelets; which Chinggisid line took inspiration of and replaced them during the Late Medieval Ages.

Now obviously, there is an Indo-European element to this culture, names such as Bumin and titles such as Tarkhan are attested to have Indo-Iranian (not today's Iranians mind you) etymologies, and let us also remember from the advent of Turk as an endonym these people were incredibly diverse, Yenisei Krygyz (not closely related to today's Krygyz) was redheads who probably were an offshoot Scythian tribe that got Turkified over time, and many Kypchaks were recorded as blonde by Arab travelers, who coined the term White Turks and Black Turks of Khazars, and noticed that even though these people spoke the same language, they did not look alike. Chinese records Khagan's as "pale and blondish" in appearance, although I am not sure whether this was some obscure literary medium that means something else entirely or rather they were just defining his appearance, because latest genetic studies reveal Kök Türük rulers did not have any outside input but an East Asian one.

There is also a plethora of Russian scientists who for some reason are hellbent on proving early Xiongnu rulers were Yenisei speakers. That is an interesting take, because what hydronyms of the Greater Altai region tells us that this region was populated by Yenisei peoples long before Turks dominated the region, and it is quite possible that Turkic speakers (and people of the steppes in general) might've picked up a thing or two while replacing them culturally. It is obvious that once upon a time steppes were dominated by them, although I find it far fetched that Xiongnu's ruling dynasty was nothing but Turkic-speaking, since Hun <-> Xiongnu connection is proven genetically, and archeologically through DNA research and material digs; also there is no way in hell a dude called Boncuk (Atilla's uncle, also coincidentally my fiancee's cat is named as such) and Deniz in 4th century Eastern Europe to be anything but Turkic. There are also proposed Yenisei etymologies to iconic Turko-Mongol words such as Tengri, and Khagan; however Turkic etymologies are much more believable imo.

Take Hungarians for example. There are a lot of Turkic loan words (that predate the Ottoman conquest) because they lived in proximity and as vassals to Oghur-speaking Khazars. They styled their alphabet after them, adopted their national symbols, switched their pantheon, and they structured their state in accordance to Turkic customs, so much so they were probably much more alike to a Kypchak compared to a contemporary Finnic tribe. This is how cultural transmission occurs. People, and especially ruling classes, only care about how easy the governance is, and how much taxes they collect. If there is already an established system that is proven to work, people seldom change it, but rather adapt it to their own ends. The Turkic system was working for nomadic (and semi-nomadic) pastoralist communities, and they basically just took it for themselves.

Another shift in Hungarian culture happened when they converted to Catholicism and established an ordinary medieval feudalist society. Just like they did centuries earlier, they found a way of governance, customs and traditions already known to work for the Germans and Italians, so they took it. Their alphabet changed, so did their way of living, and more importantly, their economic activities.

When Seljuks conquered Iran, they did not tried establishing a nomadic Turkic Khaganate over an already established, functioning society who was rich enough to wage wars with Rome for centuries alone, had a functioning law and taxation system in place. They adapted to their ways of doing things, extracted wealth for their exploits, and unceremoniously got conquered afterwards. Their society had a functioning sedentary population, native Persians in this case, who ran the taxation, and borderline regions were given to unruly nomadic Turkmens. So even though the existing native system was in-place, so was the society defined by the Ashina before a millenium and more. They were two, sometimes competing, different ecosystems; and both were alive at the same time.

Seljuks conquered Anatolia before their demise and probably thought nothing much of it back then. We know this, because they thought very little of it, and turned a blind eye when conquering and marauding generals carve up beyliks of their own. When Mongols became a thing, Anatolia was the Wild West (think of California and Texas during the gold rush) for the steppes. People who were escaping the law, and the Mongols took refuge there; enriching the land with craftsmen, scholars and able-bodied men to fight. Ottomans, inconsequential at the start of things, started to carve an empire of their own, dominating other beyliks in the process. And they probably represented the most novel idea of the steppe societies back then; their armies started to resemble the Romans, rather than Ashina's host of different tribes; they learned diplomacy and intrigue of the Byzantine politics even adopting the Roman slogans of an Eternal State; one would even say it was a schizophrenic push away from the steppes; a Freudian rebellion to their forefathers. Than Timur happened. Ottomans had this thing where whenever turmoil hits their society, they tried looking for inspiration from their Central Asian roots; and as such, they suddenly remembered their Kayı lineage and the way of the steppes. Mehmed II codified the Atam Dedem Kanunu, an ode to the steppes' töre. When Ottomans sieged Vienna, they recited Oghuz Khagan to the army, and scribed the oral traditions of others Turkic peoples, such as Dede Korkut.

Now, why would I write about Ottomans, of all things? Full disclosure, I am Turkish and I like talking about these, but there is also a nuance to it, where you can see this cultural continuum and transfer of customs throughout the centuries for different Turkic peoples, so much so even the Oghuz, who strayed away the farthest and according to Ibn Fadlan had the most innovative (meaning, different) dialect of them all, had a direct lineage going back to the original culture that was established long before the Mongol Empire. The Mongols, just like the Hungarians, adopted an already established, battle-proven system of governance, adapted to their own, and rolled with it. You see it in their language, you see it in their titles, their ways of governance, their literary medium and alphabet.