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Let's take a quick snap at what really happened to the branches of the Mongol Kingdom;

The Golden Horde transformed into a Qibchaq-Turk state, and then states. They started to speak Turkish as first language and gradually gave up the Genghis Law.

The Il Khans, though resisted in their Mongol identity for longer, devolved into local Turkic/Persian dynasties.

Kubilai'i descendants became the Yuan dynasty of China, Chinese in all respects but blood.

The Mongol hordes located in Western Turkestan, though preserved the Genghis Law for longer than any, adopted Turkish dialects and amalgamated with the urks and other locl population to give way to the folks of modern peoples inhabiting there. The outblow of Tamerlane was an offspring of the Chaghatai branch but already speaking Turkish...

In summary, wherever they went, they gradually dissimilated into the local cultures/populations and became the integral elements of their new habitats, no more retaining their initial identities but contributing many elements of it. They had to, because they were always a minority (population-wise) wherever they invaded.

Therefore, little would have changed if they continued their march westwards or in other directions. They might have flamed for longer, but would eventually die out like they did.
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