Depopulation and movements of population by varying degrees of force and compunction did happen throughout the 30YW and 80YW. However, this is a difficult question to answer precisely because (1) 20-year rule, (2) estimating the degree of population upheavals in that era is notoriously difficult.
To give an example, we may be able to say that the city of Magdeburg declined from 30,000 to 5,000 due to the Sack of Magdeburg, an infamous example of cruelty in the 30YW. But it is very hard to say what happened to the 25,000 "missing" residents, whether they were killed or whether they fled and tried to re-settle elsewhere.
But there is at least one well-known instance of a large group of people who displaced themselves by choice, namely the Calvinist and Jewish burghers of Antwerp, who moved out of the city following the Duke of Parma's success in taking it. Following the surrender of Antwerp to the forces of Philip II of Spain in 1585, the Duke of Parma promulgated a policy of tolerance, as he hoped for Antwerp to once again rise to become an important economic center of the Low Countries. He didn't want a repeat of the infamous Sack of Antwerp, thus the Protestants were given four years to settle their affairs before they have to leave or convert, the hope being that some may convert. Instead, a great many moved themselves to Rotterdam and Amsterdam. Estimates of population give numbers around 100,000 prior to the fall of Antwerp, and around 40,000 four years afterward.
At the same time, it is important to note that Rotterdam, Amsterdam, and other cities of the Dutch rebels strongly sought after these burghers and their talent and their capital. I shall end here lest I run into the 20-year rule.
What comes to mind for me is the so-called Migration Period during late antiquity, into the Dark Ages. Germanic peoples, under pressure from their own resource scarcity and the encroachment of powerful neighbors further to the East, crossed the borders of the Roman Empire with the intent to settle.
Most readily the example I can give is that of the Visigoths. In 376 AD, the Goths asked the Emperor Valens for permission to cross the Danube and settle. The Emperor's permission given, the Goths relocated and were then treated poorly by the Romans, leading to a rebellion that is quite another story in and of itself. (Valens died for his hubris)
But throughout early European history there are example of large groups of people migrating to escape war, or resource scarcity. The Cimbri & Teutons of Gaius Marius' day, the Helvetii and Suebi of Caesar's. Later on the Magyars would launch an invasion of the central European plains and found what we now call Hungary.
It's true many of these migrations took on a martial character, but that shouldn't detract from the fact that many of these populations were more or less refugees, and moved women and children along with their men. My own expertise tends more towards the Roman experience of migratory peoples, so that's what I've focused on.
Some incursions were more strictly martial in nature, with an aim of conquest. But others were different. The Visigoths have every appearance of wanting to make a good faith treaty where they'd become Roman subjects but their shoddy treatment afterwards led them to rebellion. Likewise the Helvetii showed every appearance of wanting to migrate to wherever they could and not bother the Romans, but Caesar either didn't trust them or simply wanted the glory of defeating a Barbarian Horde.
But in any case, yes there have most certainly been events that might be called 'refugee crises' where large populations and even entire nations of people have tried to transfer themselves across national borders. Due to the attitude of the Romans, and the circumstances of the times, these often led to bloody confrontation in contrast to today's circumstances.
One last point, the most recent example I can think of is much closer to us than the Romans, but I assume you'd be interested in it by virtue of being 70 years ago, although it may not fit the textbook definition of 'Pre-Modern'.
I'm referring to the mass population exchanges in the aftermath of World War 2. Largely compulsory, many ethnicities were shifted around to help match and sustain new borders - millions of ethnic Germans being deported from Poland, the Baltic, and Prussia into the redrawn Germany. Likewise Poles pushed westward, out of the areas the Soviets annexed from the old Polish Republic.