The Kalmyks were a group of Oriats who migrated from Dzhungaria in 1607 (e.g., several hundred years after the Mongol Khanate). While they migrated into the Astrakhan region against the wishes of Russia, they remained a "state within a state" at Russian suffrage, because they were a useful ally against the Muslim Turkic peoples who otherwise occupied the area.
It's hard to imagine the Kalmyks migrating much further west, as neither the Ottomans nor Poland-Lithuania had much reason to accept them into their territory.
It's also worth noting that prior to the 17th century, most Mongols were not Buddhist, so even if you managed to get a "Mongol" population into Europe, it would likely not be Buddhist, but probably either Christian or Muslim. Basically a racially mixed population similar to the Hazara in Afghanistan.