They were originally another Chinese ethnicity several thousand years ago, and remained so throughout time.
Their "ancient" records were recently translated by Russian archaeologists ~ who were extremely curious about this very strange group who speak an anomalous sort of Turcic language.
It looks like these are the people who traveled North and South from Siberia to the Ganges valley on immensely long treks to trade all sorts of things. Think of their route as like a piece of The Silk Road with an Eastern leg to Nepal and India.
BTW, people have always been curious about why the Sakha herded cattle and not just the more traditional reindeer, yaks, goats, horses and sheep. Nobody else in that region of the world herds cattle.
Now we know.
Apparently they made a pretty good living with their way of life and could afford to keep up with the Chinese (nearby) when it came to military technology.
They arrived in Korea and Japan some time in the mid 500s AD. This would be a result of being driven out of their Siberian rangeland by the climate anomaly that created the Dark Ages in Europe and Northern Asia and China.
The reason folks know when they arrived in the Far East was simple ~ they began a long term conquest of those territories ~ with Japan taking the longest. War didn't stop until the Tokugawa Shogunate, that that's about 1000 years.
The Saka are mentioned in the Mahabarat as taking part in the War between Truth and Lies.
Most of the other meaningful Turkic invasions of India have been undertaken by people East of the Himalayas.
Buddha was a Shakya, not a Sakha. Shakya was one of the numerous janapadas in northern india, and the people were Kshatriya. This is quite different from the Turkic Yakuts who didn’t move from their areas around Lake Baikal southwards until well into the 10th century AD