HISTORY:

DZUNGAR KHANATE
ORIGINS
The ancestry of the Oirats is from Naiman, Kerait and Merkit stock. However the first Oirats were a group that emerged after the fall of the Yuan Dynasty in China and began the long feud against Halh Mongolia for the next 400 years.
TERM
Oirat comes from the term Dörben Oirat also known as the "Alliance of Four". It started with four western Mongolian tribes Khoshut, Choros, Dörbet and Torghut, then incorporated other tribes such as the Zachachin, Bayids and Uriankhai.
EARLY HISTORY
Oirats were never fully united into a single power for the longest time. Toghon Tayisi in the 15th century expanded Oirat rule greatly but after his son Esen Tayisi (Who managed to capture a Ming emperor) was slain in civil war - over his diplomatic screwup with the Ming. The Oirats fell into disunity and fought amongst themselves.
RISE OF THE DZUNGARS
In 1606 however, Khara Khula emerged to unite the Oirats against the Halhas who was attempting to annex Oirat territory. He himself was descendant of Esen, and formed the Dzungar nation. He didn't call himself a Khan however, and was strictly a military ruler. In 1609 he won a decisive victory against the Halhs and established sovereignity over Dzungaria, but it wasn't until 1623 that Ubashi Khong Tayiji - Khan of the Altan Mongols was killed and thus established Dzungar independence from the Mongols.
Dzungars did not have a Khaan until 1671 by Galdan Khaan. It was during this period that the Dzungar Khanate came to be as it is famously known - territories owned were Dzungaria, East Kazakhstan, and Kyrgyzstan by 1679.
PAN-MONGOL CONQUESTS
As Manchuria grew in power to the East however, Galdan Khaan wanted to re-unite the Mongols by invading Mongolia in 1688. He was backstabbed by his nephew Tsewang Rabtan who usurped his throne in 1689 however.
Dzungar armies advanced all the way to Eastern Mongolia, then proceeded south as the Halh royalty fled to inner-Mongolia - who then surrendered themselves to the Manchus for their support against Dzungar armies. The Qing Emperor agreed, and sent a letter to Galdan asking for peace - however, it was not peace he had intended.
The battle of Ulaan Budan was no battle. It was an ambush, crushing Galdan's unprepared forces and forced him back to Halh Mongolia. Galdan was isolated from Dzungaria (which his nephew usurped) and could not recover from this defeat. By 1696 a combined force of Halh Mongols and Manchus invaded Mongolia and encircled Galdan's remaining forces at the battle of Zuunmod. Queen Anu sacrificed her life to save Galdan from encirclement in which he survived - but killed himself in 1697 to avoid capture.
AFTERMATH
The Dzungars were completely in disarray at this time fighting amongst themselves over succession. It was completely lawless, and its people were also inflicted with the smallpox epidemic which spread across the region at that time. It was during this time when Prince Amursana also betrayed the Dzungars by joining with the Manchus on the condition that he would be called Khaan. He had his own plans however but was found out somehow and fled to the Russians for refuge.
GENOCIDE
In 1755, Cebdenjab, son of a Halh Mongol prince, led the expedition for the Qing Emperor and exterminated the remainder of the Dzungar people who did not migrate westwards towards the Caucasus. It was genocide, numbers are unknown - it is unknown also whether the majority of the population were killed due to small-pox or by the expedition. All were massacred, including men, women, and children in Dzungaria. After the final Khan Dawa Achi was slain in 1759 that the massacre ended. The Manchu Emperor repopulated the territory then called it Xinyiang - or "New territory".
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MAP
Here's a map of the Dzungar Khanate at 1720 (23 years after Galdan's death - and rule of his nephew):

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KALMYK KHANATE
There is a rather long history of the Kalmyk Khanate in the Caucasus. I really can't be stuffed putting it all together. But here's a general history:
MIGRATION
Kalmyks - means 'remnant'. The branch of Kalmyks existed in the Caucasus before 1755 - as the map above from Russian sources shows. After the war against Halha Mongolia and Manchurian Empire in addition to the smallpox epidemic where most of our men, women and children were exterminated, the remainder of us migrated westward. Why extermination - which was not the nomadic way? Ask those Halha buddhists. Yellow sect buddhism is as theocratic as any other crazy religious sect.
RISE OF KALMYKIA
We ended up evicting the native Turks of the caucasus, then we aligned ourselves with the Russian Tsars to fight against other Turks, Muslims, and Ottomans. We were doing great until Ayuka Khan died, then the Russians betrayed us and we ended up taking them on while at the same time every other Turk hated us.
RUSSIAN OCCUPATION
In the end Russia incorporated us into their Empire. Centuries down the track we fought for both Nazi Germany (against U.S.S.R.) as well as for the U.S.S.R. There is a great memorial statue crafted in our honor in defence of the motherland.
However, due to many Kalmyk guerillas (who joined the Nazis) giving the Russians so much trouble, Stalin had the nerve to decide "hmmm, Kalmyks, many Kalmyks joined the Nazis and killed a lot of Russians... no can't trust them, I'm going to deport them".
GENOCIDE (the 2nd - heh as if the Jews can even claim this)
The Russians deported us all over the place in 1943 and many died because of it. Heroes who fought for Stalin - returned home, and found out that our people were being deported. So much for a reward! We had no food, no clothing, nothing, most of us died during the deportations. Betrayed AGAIN!
REFORMATION OF KALMYKIA
1957 was when we were allowed to return home. The area was repopulated by Russians and Ukrainians however, and they wished to stayed. So we didn't really have much choice. However, Kalmyks now have their own autonomous state inside the Russian Republic.
Dzungar/Oirat/Kalmyk has many famous people under their banner and heritage - Vladimir Lenin himself, General Subeedei himself from the Uriankhai tribe, and many others.