My understanding is that any sort of a census of the sort described in Luke (requiring males to return to the towns of their birth for registration) probably did not actually occur, as it would have created impractical logistical issues.

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https://en.wikipedia.org/wiki/Census_of_Quirinius

Most critical biblical scholars agree that the Gospel of Luke is erroneous.[10] Its author seems to have invoked the census as Joseph and Mary's motivation for departing "their own city"[11] of Nazareth, Galilee, for Bethlehem.[12][13][14] Additionally, the author may have wished to contrast Joseph and Mary's obedience to the Roman edict with the rebelliousness of the Zealots, and also to find a prophetic fulfilment of Psalm 87:6: "In the census of the peoples, [princes] will be born there."[12][b][c] (Luke and Matthew also give different accounts of the family's departure from Bethlehem.)[17][d] Catholic priest and biblical scholar Joseph Fitzmyer states:[20]

It is clear that the census is a purely literary device used by him to associate Mary and Joseph, residents of Nazareth, with Bethlehem, the town of David, because he knows of a tradition, also attested in Matthew 2, that Jesus was also born in Bethlehem. He is also aware of a tradition about the birth of Jesus in the days of Herod, as is Matthew; Luke's form of the tradition, unlike Matthew's, tied the birth in a vague way to a time of political disturbance associated with a census.

Scholars point out that there was no single census of the entire Roman Empire under Augustus and the Romans did not directly tax client kingdoms; further, no Roman census required that people travel from their own homes to those of their ancestors. A census of Judaea would not have affected Joseph and his family, who lived in Galilee under a different ruler; the revolt of Judas of Galilee suggests that Rome's direct taxation of Judaea was new at the time.[21] Catholic priest and biblical scholar Raymond E. Brown postulates that Judas's place of origin may have led the author of Luke to think that Galilee was subject to the census.[22][e] Brown also points out that in the Acts of the Apostles, Luke the Evangelist (the traditional author of both books) dates Judas's census-incited revolt as following the rebellion of Theudas, which took place four decades later.[22][f][g] Protestant theologian and biblical scholar E. P. Sanders concedes that, because revolts followed both Herod's death and the census, Luke's author may have "decided to elaborate the event [as] a reason for Joseph to travel ... to Bethlehem", calling this "a relatively slight historical error for an ancient author".[8]