With refugees now trying to get out of the war-torn Middle East or out of Africa into Europe, there is often a strong reaction, Europe for the Europeans. But in fact Europe has witnessed waves of migration, immigration, and downright hostile invasions throughout recorded history. The ancestors of many modern day Europeans (and their American descendants) were migrants or invaders in their day.
Take England. There were the megalith builders thousands of years ago, heaving Stonehenge and many other huge stone structures into position (while Neolithic Woman doubtless rolled her eyes and wondered when Neolithic Man was going to go hunting and get the family some food for a change). Then the Celts came over the land bridge that then connected the British Isles with the Continent. Then the Romans came and conquered (first century BC). Then the Angles and Saxons came and conquered (fifth-sixth centuries AD). Then the Normans came and conquered (1066). Modern England is a mix of all of these, and it would probably have a lot more in the mix if the chalk land bridge hadn't been broken through (well BC), creating the Channel.
In the early Middle Ages, there were plenty of migrants, refugees, and invaders. The Germanic peoples for the most part came peacefully into the Roman Empire and its margins. The Franks settled in Roman Gaul and quickly adopted Roman culture and language. The ancestors of the Scandinavians settled in the northern parts of Europe where the Romans had never gone, as the ancestors of the Swiss settled, with their cows, on mountains the Romans had avoided.
Other Germanic peoples conquered, like the Goths who sacked Rome in 410. (This was of course a major shock to the Romans, but they rebuilt--this did not cause any "fall.") But the Goths too ended up settling in Italy and Spain and adopting Roman ways, being indeed recognized as part of the Empire.
Then there were the Huns, who rampaged through Europe in the middle of the fifth century, being stopped from sacking Rome by the pope (Leo I), and whose empire collapsed after the death of Attila in 453. The Magyars, a related people, ravaged the German kingdom five centuries later, before being stopped by the German king; modern-day Hungary considers itself the heir of both Huns and Magyars.
The Vikings were another terrifying group of invaders, from whom many people in both France and England had to flee--although some Vikings settled in Normandy and some in Yorkshire around the beginning of the tenth century, where they quickly became respectively French and English (but with some of their lively nature still in place).
Any of these invasions of course created refugees. So did natural disasters like the famines of the fourteenth century, which sent waves of desperate people across Europe, seeking for food. Even without massive famine, local loss of crops would lead to refugees, who took off hoping to find some place that still had something to eat.
And then there were just migrants. Europe's cities grew quickly in the eleventh and especially twelfth centuries, as people (the majority young men) gave up farming to move to town for what they considered a better life. Just as in nineteenth-century America, the migration of people into the cities represented a major population movement.
For much of the High Middle Ages (after the great waves of invasions were over), refugees were treated with pity and concern. Bishops reminded parishioners of Jesus's saying, "I was a stranger, and you took me in." This only lasted as long as the economy was strong. With the weakening economy of the fourteenth century and the accompanying famines (and then Black Death), people lost all pity for refugees, who were considered dangerous and frightening.