There is no evidence of such an occurrence. If anything, the "Phoenician" population of the West seems to decline from the 8th to the 6th centuries BCE, though Carthage began quite large and remained so. Despite nearly constant Assyrian and Babylonian incursions into the Levant during this period, the Phoenician metropoleis never suffered catastrophic destruction. They were simply under this or that overlord, and even Tyre's frequent rebellions did not bring catastrophic consequences.

The Assyrian Tiglath Pileser III was receiving tribute from the Phoenician cities in the second half of the 8th century, precisely when the westward expansion was in its heyday. The two are not necessarily connected, in my opinion. His successors Shalmaneser V and Sargon II both had to deal with large-scale rebellions (including Phoenicians), instigated by the Egyptians. Their attempt failed, but Sargon II apparently did not bother to punish the coastal cities of Phoenicia (Damascus was probably not so lucky). Sennacherib dealt with much the same situation, apparently. This brings us down to 681 and the accession of Esarhaddon. He gave Phoenicia a little more attention, and we have some nice evidence suggesting he punished Tyre for centuries of rebellion. Here is a little bit of a record of the "treaty" between Esarhaddon and Ba'al, king of Tyre (not to be confused with Ba'al the god). Even at this point, there was probably no event to cause a mass of westward-sailing refugees.

Incredibly, Tyre was back at it again by the time the next (and final) Assyrian ruler took over: Ashurbanipal (669-627 BCE). About 7 years into his reign, in ca.662, Ashurbanipal besieged the city and somehow forced another treaty (it would last about a decade only). I think a key issue here is that Tyre was on an island in the bay, and was nearly impregnable until Alexander the Great came along. The Assyrians could besiege her, but with her extensive sea connections to Cyprus, the Aegean, and the West, it was probably impossible to accomplish much.

Anyway, from there the Babylonians take over the job of trying to police the Phoenicians, and then the Persians.