The Late Roman Empire, infamously perhaps, had to deal with groups of barbarians such as the Visigoths, Ostrogoths, Burgundians, Vandals, and Franks crossing over into Roman territory. What these movements looked like varied from large-scale migrations to smaller elite transfers, to cultural diffusion based around the Roma army. Many of those barbarian groups, or at least portions of them, were eventually settled in Roman territory via the Hospitalitas system. The sources for that settlement mention that these groups were granted "thirds", but "thirds" of what? The old idea was that they were granted bits of land, hence why we don't really read too much about barbarians taking over Roman land by force, at least not until the mid-fifth century. However, the work of scholars like Walter Goffart have put forward an interesting, and generally accepted argument, that these "thirds" refer not to land but to tax shares, and that the method of settlement then was initially a tax system to pay the barbarians working in in the late Roman military. Eventually, this system of hospitalitas enabled the successor kingdoms of early medieval europe to form out of the settlements of the late roman army.
SOURCES:
Barbarian Migrations & the Roman West, Halsall
The Inheritance of Rome, Wickham
barbarian Tides, Goffart
barbarians & Romans, Goffart
Empires & Barbarians, Heather
Early Medieval Europe, Collins