Well for the purposes of this, we'll say that Asia will be the India, China and Japan. The Mongols could well be the Russians of this scenario.
And the idea would sort of work like with the British in India. It starts with trading posts, then getting involved in local wars and before you know it your administrating the place.
@Kalan
Your 3) and 4) are the ones I'm aiming at.
The Asians are not in Europe for any specific resources, though amber is desired by the rich folks of China and Northern India. They are in Europe to sell stuff rather than buy.
In that case the closest I could come up with is a further Han Dynasty or Tang Dynasty push west where they establish some trading post somewhere on the Black Sea coast. Perhaps silk for glass. I can't see this limited overland trade being economically sustained. But it would be interesting to see how China and Rome/Byzantine interacts.
By far more likely would be a Mongol settlement in Hungary, if Mongols fit your definition.