I have read that Mughals were used to Persian traditions. They used Persian language as the language of administration under their rule. I have also read that they were the descendants of both Timur, who was a Turko-Mongol, and Genghis Khan, who was a Mongol. So, why did they follow Persian culture if they were not related to Persia at all?
First, the relationship was
Mughal Empire -> Timur Empire -> Mongol
; I mean that while the Mughal Empire had distant Turco-Mongol origins, its most direct influence was that of the Timurid Empire.
It was under the Timurid Empire that the Turco-Mongols adapted to the Persian culture (and passed that tradition to the Mughal Empire). The Timurid Empire adopted that culture when it was under control of Persia.
Why did the Timurids change? Because:
they were a relatively small military elite ruling over a big population. It was in their own interest to "align" themselves with their subjects to avoid motives for rebellion.
they were a nomadic culture who suddenly became the rulers of a sedentary civilization. Managing a sedentary civilization -with its cities, its farming, etc.- required a different mindset/tradition that the ones the Mongols had, and presented a new set of problems. The Persian culture and traditions were already stablished, were adapted to that environment, and solved those problems.
the Timurids had conquered Persia because it was a rich, profitable objective as it was. In order to profit from that wealth, it was easier to leave the Persian with their old traditions and just become their leaders than converting the Persians into Mongols (which also risked to simply make the Persian country as poor as the Mongol country had been). The Timurids were happy to "let things stay that way" and just collect the profits.
All of this caused the Timurids to become progressively more and more "Persianized", and that tradition was transmitted to the Mughal Empire.
The above explained process was far from unique; almost everytime a nomad society conquered a civilized one, it ended with the nomad society adopting the culture of the conquered society. The Mongols alone offer plenty of examples: the Timurids in Persia, the Golden Horde in Russia, the Yuan dinasty in China, all became "local" very quickly. The Germanic invasion of the Roman Empire is another classical example.
Even when the difference of civilization level was not as important, in situations of a small force dominating a big population of a different culture, many times the invading force became very influenced by the local population (Macedonian rulers of Persia and Egypt, Crusader kingdoms, etc.)
Well, adopting Persian was more than just about a language. It gave access to a whole civilizational complex including culture, literature, governmental techniques, etc.