Over 50 million children today are refugees. Even if Jesus had not been a refugee during his flight to Egypt, He is those refugees now.
When the worldwide refugee crisis was really understood in Europe and the United States, Christian and non-Christian commentators began drawing comparisons between refugees and the Holy Family, especially at Christmas. In Matthew 2, Mary, Joseph, and Jesus were forced to flee from King Herod, who sought to kill Jesus, to Egypt. On the surface, this appears cut and dried. Fleeing the king in Judea, Jesus went to Egypt for safety. However, arguments to the contrary quickly gained traction, arguing that Jesus never crossed an international border.
"Jesus never left the land of his sovereign—his move between parts of the Roman Empire is roughly analogous to moves between Alabama and Florida in the US or Portugal and Germany in the European Union." (Capstone Report) "The Holy Family traveled from one province of Rome into another to escape persecution. As a matter of political organization, such a voyage more closely resembles the flight of overtaxed Californians into Texas than it does a caravan of Hondurans into San Diego." (Michael Knowless)
By this argument, the Roman Empire was much like the United States. Judea and Egypt were like US states, and Rome was like the federal government. They never left the equivalent of the USA, and so, since the 1951 Refugee Convention defines a refugee as someone who has crossed a national border, Jesus could not have been a refugee.
But is this the case? Judea, Egypt, and other provinces functioned as vassals in the ancient world, paying tribute under the power of Rome with various levels of autonomy. Before Late Antiquity, empires had terrible centralization problems in conquered nations, and they delegated most functions in regions which were not too problematic to local leaders who paid them tribute and bowed to their wishes. Because of this, the exact nature of the relationship between provincial power and empire power fluctuated. This is why, for instance, the Jews could return to the Levant after the Exile and claim to be "the kingdom of Judah" with their own law code and government system, but Persia claimed them as part of the empire.
So why compare the Holy Family's flight to changing US states? Why not compare someone who remains in Syria but flees from Assad or ISIL control to Kurdish control, which would not cross an official border but instead jurisdictional control? Or perhaps someone who flees from an Israeli settlement to Israel proper, remaining within one jurisdictional government but cross an official border?
When Israel built the West Bank border wall, a few Palestinians were left on the Israeli side of the border wall and the new border. Did they suddenly become refugees without ever leaving their homes? During the Sudanese war, many people fled south and settled there. When South Sudan was founded, did they suddenly become refugees because a new country was formed? When ethnic Germans Soviets crossed a national border from the oncoming Soviet advance, did they cease to be refugees once East Germany became under Soviet control? These are hardly perfect or even particularly good comparisons, but it does illustrate how requiring a refugee to cross a "border" becomes fuzzy. The fact of the matter is, Egypt and Judea were conquered states which maintained jurisdictional autonomy separate from each other. The Holy Family could flee the ruler of Judea, a conquered nation with autonomy, to Egypt, a separate conquered nation with autonomy. So yes, in any international law court, the Holy Family would be understood as refugees because of these border ambiguities.
But these arguments over legal definition and Jesus' flight as a child miss something. Even if Jesus was not a refugee as a child, He was, is, and will be a refugee. In Matthew 25, Jesus says He was a "stranger" who was either invited in or rejected. The refugee is Jesus, just as much as the hungry or thirsty or imprisoned. The Greek root for this word is also used in the Septuagint. In 2 Maccabees 5:9 it describes someone who had been "driven from their own country into exile and died in exile." It's also used in 2 Samuel 15:19, "You are a foreigner, an exile from your homeland." It is also Ruth's identity in Ruth 2:10.
Whether or not Jesus fits as a refugee in his flight to Egypt, refugees are part of the Son of Man's judgement in Matthew 25. Over 50 million children today are refugees. Even if Jesus had not been a refugee then, He is those refugees now. If we do not accept refugees with hospitality and comfort, we are in violation of God's justice. Jesus is the asylum seeker at the US border and child on a beach in Turkey, and we are responsible for their welfare.