Strictly speaking, I don't think there was any large-scale population movement around 476 when Romulus Augustulus was deposed. By then, the Western Roman Empire only really controlled Italy and the regime that replaced it, that of Odoacer, was a highly Romanised one, so I don't see why people would flee anyway, since the transition was relatively peaceful. The new king in town had served as a Roman general and preserved Roman institutions, no matter how powerless, such as the Senate; he even sought recognition from Constantinople, so he was able to portray himself as the rightful ruler of Italy under the suzerainty of the rightful emperor. After a brief conflict, Theodoric's Ostrogothic kingdom that succeeded Odoacer's regime was similarly Romanised and his kingdom may well have been seen as a renewed Western Roman Empire by his subjects, or it was just propaganda. Either way, Theodoric's regime was stable until its very end in the 520s when religious persecution picked up, but even then we have no evidence that his purge extended beyond the Senatorial aristocracy.

There were refugees from the west in the fifth century, but they did not flee uniformly to the east nor did they do so at the same time. We know for example that after the Vandal conquest of North Africa in 439 many landowners fled to both Italy and the east, to the extent that after the Eastern Romans reconquered the province in 534 estates confiscated by the Vandals were returned to their descendants. Over the next few decades, we also hear of Nicene Christians fleeing religious persecution under the Arian Vandals, but as before, we can't quantify it; in any case, the persecution was sporadic, so the number of refugees must have fluctuated over time. Another example can be derived from the Life of St Severinus, a record of a saint active in Noricum during the collapse of Roman control in that province. The author had in fact fled from Noricum after the saint's death, but he fled to Naples rather than the east - Odoacer's regime had for some reason ordered local Romans to move to Italy after he won a victory over the Rugians in that province. Evidently, though the Western Roman Empire no longer existed, central authority was still exercised by some states and they were keen to protect civilians, either out of genuine altruism or for more cynical reasons.

As for the provinces further west, I can't think of any specific examples of refugees travelling to the east. Although I'm sure many people were refugees in times of war, the collapse of the Western Roman Empire was a gradual process that took many decades to occur and it had different results in different regions. In some cases, political control was transferred to new Germanic rulers both legally and peacefully, so there would have been no reason for locals to flee. Even if there was conflict, fleeing to the east would have only been a choice for the wealthy, the section of society most likely to do well under their new rulers, who were always keen to employ highly educated literate men in their administration. Though the fifth century was indeed an age of economic decline and occasionally devastating conflict, it wasn't entirely all doom and gloom either - plenty of people did very well for themselves!