That's a great question. We can ask: how long does it take before a culture loses the memory of a migration? The answer, for a civilization without written records, seems to be "on the order of a thousand years"
How do we know that?
Consider a few examples:
Aborginal Australians- they have intensely detailed oral traditions of their land ("songlines") which record long ago geographies; but they don't have a recollection of their arrival in Australia, some 65,000 years ago. They view they themselves as "autochthonous", sprung from the place they are, not migrated from somewhere else.
Native Americans- they're similar to aboriginal Australians, with a very deep attachment to "the land", and no sense that there had ever been in any other land. So that's at 15,000 years ago; they _do_ seem to have memories of migrations on the order of one to two thousand years ago; not crossing the Bering Sea, but dispersals and relocations in the Americas-- sometimes we can match these accounts with geologically datable events.
Then we get to Maori in New Zealand/Aotearoa, who likely arrived somewhere between 1250 and 1300 CE. They have very specific stories of who arrived (a warrior named "Kupe") and where (Hokianga Harbour). They also have a name for where they came from (Hawaiki), but its not clear if this was meant to be a specific place, as the term also has a mythic connotation sort of like the Garden of Eden. They plainly did come from somewhere, but they no longer recall.
Now here's what's interesting: there are a few Maori iwi (more or less what an American would call a "tribe" or a Canadian would call a "band") living on East Cape, who don't believe the Kupe arrival story-- they think, like aboriginal Australians and native Americans, that they're autochthonous, that they've always been just where they are.
Or the Roma ("gypsies") -- genetic evidence suggest that they left Northwest India sometime around the year 1000 CE. Although the Roma have many stories of their travels, they don't have a specific story of this origin.
That's admittedly not a whole lot of data, but the Maori and Romani data suggest that an oral tradition of migration may be lost after something on the order of a thousand years. Because the Pacific Rim has so many volcanoes, which can be precisely dated, we have another yardstick for judging "just how far back do these stories go?" All around the Pacific, its common to have stories of eruptions within the last one to two thousand years-- but they get rarer as it get older than that; still, particularly in Australia, there are occasionally stories that can be construed to point back much further than that.
Sources:
The Story of Kupe. As Written Down by Himiona Kaamira
Aboriginal Histories, Aboriginal Myths: An Introduction
Fished up or Thrown down: The Geography of Pacific Island Origin Myths
Origins, admixture and founder lineages in European Roma
and see particularly
Northern Athapaskan Oral Traditions and the White River Volcano
for a systematic attempt to identify traditional stories with historical volcanic events, at what appears to be the edge of memory-- that is, there's a vague reference an awareness of volcanism that occurred 720 CE, but the specificity has been lost. In another thousand years of oral transmission, it's reasonable to assume that the story might have disappeared completely, or be so vague as to be unrecognizable.