People drifted toward the empiure, aggressively, speculatively, or desperately, because of the perception that the empire was wealthy, which is more or less the same as today, albeit that aggression cannot compete with Europes military strength aside from terrorism. The classic example in antiquity was the arrival of the Goths. Fritigern, who had already made a peace treaty with Valens, asked the Roman emperor for permission to cross the Danube and settle in Roman territory. Valens allowed this, but the governors of Thrace, where the refugees were crossing the river, were not best pleased at having to support a migrant ppulation. So Lupicinius and Maximus encouraged the most shameful exploitation of the Goths.
When the barbarians who had been conducted across the river were in great distress from want of provisions, those detested generals conceived the idea of a most disgraceful traffic: and having collected hounds from all quarters with the most insatiable rapacity, they exchanged them for an equal number of slaves, among whom were several sons of men of noble birth. Ammianus Marcellinus
Soon famine and want came upon them, as often happens to a people not yet well settled in a country. Their princes and the leaders who ruled them in place of kings, that is Fritigern, Alatheus and Safrac, began to lament the plight of their army and begged Lupicinus and Maximus, the Roman commanders, to open a market. But to what will not the "cursed lust for gold" compel men to assent? The generals, swayed by avarice, sold them at a high price not only the flesh of sheep and oxen, but even the carcasses of dogs and unclean animals, so that a slave would be bartered for a loaf of bread or ten pounds of meat. When their goods and chattels failed, the greedy trader demanded their sons in return for the necessities of life. And the parents consented even to this, in order to provide for the safety of their children, arguing that it was better to lose liberty than life; and indeed it is better that one be sold, if he will be mercifully fed, than that he should be kept free only to die. Jordanes
Not content with this activity, the two Roman governors perceived there might be trouble ahead and so hatched a plot...
Now it came to pass in that troubIous time that Lupicinus, the Roman general, invited Fritigern, a chieftain of the Goths, to a feast and, as the event revealed, devised a plot against him. But Fritigern, thinking no evil, came to the feast with a few followers. While he was dining in the praetorium he heard the dying cries of his ill-fated men, for, by order of the general, the soldiers were slaying his companions who were shut up in another part of the house. The loud cries of the dying fell upon ears already suspicious, and Fritigern at once perceived the treacherous trick. He drew his sword and with great courage dashed quickly from the banqueting-hall, rescued his men from their threatening doom and incited them to slay the Romans. Thus these valiant men gained the chance they had longed for--to be free to die in battle rather than to perish of hunger--and immediately took arms to kill the generals Lupicinus and Maximus. Jordanes
And so a refugee crisis became a full blown rebellion against Rome.