Pretty much. Mongols aside successive waves of steppe people went west to raid Europe after they got their asses handed to them by Chinese dynasties. Or sometimes it's even more indirect than that, sometimes the Chinese kick out a horde, and they fight someone in central asia and kick them out, and that group end up going to Europe to cause trouble.A bit off topic, But doesn't China has the best track record against Martial/Nomadic people when compared to other Mercantile/Settled farming people, All contemporary ancient Civilizations lost their ground to "barbarians", Sumer lost to Akkadians Indus Valley to Indo-Aryans Egypt to Rome and Rome itself to Germanic tribes but China still stands they even put up a 23 year long resistance against Mongols the Martial race that wiped out the best European and Arab armies of that time and made the biggest empire prior to Industrialization, I mean you can get more martial than that.
Imagine you were an European back in those days and you ask one of those "horse people" why they're here and they tell you an empire in the far east kicked them out of their homeland. That's a pretty scary thought.