AHC: Mongol and/or Buddhist State in Europe?

So, I was reading about Kalmykia, and it got me thinking - is it possible that, under any circumstances, a state closer to Europe (i.e. no further east than Belarus, I guess) could have been formed out of eastern conquests of the west and persist as a Buddhist state until, basically, the modern day?
 
Historically, there were several instances of practitioners of Buddhism traveling to Europe in the Greco-Roman era. Perhaps it could serve as a semi-replacement for Christianity if you can get such practitioners to hang around and get rid of Jesus?
 
Maybe the Kalmyks migrate in large enough numbers to settle in the Wild Fields of Ukraine (and probably conquer the Crimean Khanate?). Other than that, any answer probably means an offshoot of the Golden Horde does the same thing (settle in the Wild Fields) but ends up Buddhist rather than Muslim. They'd need protection from both Muslim states (Ottomans, other Golden Horde offshoots) and Christian states.

Could the Crimean Khanate somehow survive as a buffer state between Russia and the Ottomans?

But they were neither Buddhist (they were Muslims) or Mongol (they were Tatars).

Historically, there were several instances of practitioners of Buddhism traveling to Europe in the Greco-Roman era. Perhaps it could serve as a semi-replacement for Christianity if you can get such practitioners to hang around and get rid of Jesus?

This is the other option.
 
So, I was reading about Kalmykia, and it got me thinking - is it possible that, under any circumstances, a state closer to Europe (i.e. no further east than Belarus, I guess) could have been formed out of eastern conquests of the west and persist as a Buddhist state until, basically, the modern day?

Maybe a Kalmykian khanate further west, maybe in the Balkans ? Even today there are several thousand Kalmyks in belgrade.
 
The PoD must be pretty far back. Once the Hordes are staunchly Muslim it's too late. An easy one would be for Subotai to conquer all the way to the HRE, then have the Khan who rules the westernmost areas be a Buddhist who sponsors a bunch of monasteries to be built and monks and nuns to be imported from the East. It will at least establish a permanent presence of Buddhists in Eastern Europe, excepting genocide of course.
 
It would also probably help if the Mongols successfully destroyed the muscoveits and established a firm grip on Kiev, possibly making it the new Western Horde's own capitol.
 
Lots of interesting ideas, thanks everybody. I really know very little about this so you'll have to forgive me for my lack of discussion points, but it's getting the gears turning my head.
 
Kalmykia are possible because it's a semi-desert, which meant it was empty enough that the Kalmyk could take the area completely over, and Russian settlers had little reason to enter the region.
 
1 - There's a serious war between Russians and Kalmyks (insert reason here);
2 - Other European nations recognize the bravery of Kalmyk cavalrymen and they are called up to resettle in sparsely populated areas of Eastern Europe (My personal favorites are: the Austrian Banat and Swedish Ingria);
3 - Kalmyks start gaining more and more influence within their nation of choice and eventually become a plurality or even a majority in the newly settled area;
4 - Gain independence.
 
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