The site you have linked actually says Shakas originated in South Asia. You claim they were East Asians who conquered India. Nowhere does it say Buddha’s family were expelled for India. You made that up. You claimed Buddha’s family was from Yakutia. The article you pointed me to says exactly the opposite i.e Yakutias originated from India. You have clearly been proven wrong by your own source.
So, why would these guys move around that way ~ go South, then North?
Well, it's the cyclic nature of the climate. We know they also moved away from Siberia into Korea and Japan into the 6th Century AD (CE). They did that in response to the Northern Hemispheric cataclysm that brought on the Dark Ages. It wasn't mongol invasions, or barbarians from the North did that ~ it was the climate. it affected the Far East, Central Asia and Europe. In fact, Europe was reduced to sub-third-world status for nearly a thousand years. China was on its way to recovery in 300 years. The Arabs in Mecca didn't know anything had happened until they found they could conquer 1/4 the known world in a few years.
The Dark Ages were very bad in the North, and the Sakha moved to Japan and took over.
If we go back in time to an event known as the Greek Dark Ages we see an opportunity for another climate disaster in the Far East about 1200 BC to 800 BC. Iceland's Mt. Hekla erupted in 950 BC and 1159 BC. A little earlier Santorini in the Mediterranean blew up in 1630 BC giving rise to a variety of climatic dislocations for several years.
Given that the Sakha appeared to have been able to navigate around the Eastern limb of the Himalayas with ease at various times, they could very well have done this many times over the centuries, skirting China proper (and their brutal regimes) and tripping on down to India any time they wished.
We can even go so far as to suggest Chinese emperors and earlier kings actually beefed up their wall system any time the Sakha were on the move, going North or going South.
Notice the article mentions something very important ~ the Sakha genetics show no sign of Mongolian marker sequences.
These guys are NOT Mongols ~ nor are the Chinese. On the other hand, the Sakha, and the Daimyo (to the degree any of them have been sampled in these studies) have Caucasion, Indian and "other" ancestry, as well as Chinese ancestry, and due to a more recent history in Japan, they have added Jomon ancestry (the Samurai being descendants of the original Jomon settlers there).
They've probably been moving North and South to and from their Siberian homeland for tens of thousands of years, ice permitting.
Worth noting, the Eskimos and American Indians ALSO derive substantially from this same group.