In earlier times they simply road Southwest to India.
No one knows how many times they made the trips North and South and East, but they managed to maintain technological parity with the Chinese.
Further West, beyond the Himalayas, other Turkic tribes moved North and South with great regularity, into and out of India, all depending on the climate at the time.
I think the big surprise came with the Mongols who lived far to the West of Siberia rode SE to China, and then to India, and West through Russia.
That was a big one!
Genghis Khan, who led the greatest expansion of the Mongols, became Great Khan in 1206, during the Medieval Warm Period (950-1250 AD), not during a cold period.