[1] https://www.theatlantic.com/science/archive/2016/05/the-asto...
[2] https://arstechnica.com/science/2019/07/humans-may-have-reac...
Just like it happens to migrants in our time, in just a few years, not thousands.
It could be that after 600.000 years, Neanderthals got somehow behind minimal species fitness required to endure lasting species survival, and they were - this time - absorbed by the Homo Sapiens.
At looking for clues about why those initial waves went exting, maybe we could take a look at how more recent tribes or even old civilizations like Incas, Mayas, etc. made it through a couple of centuries, and eventually got "extint".
Some of those groups were surrounded by other tribes, and tried to expand their territories, with different outcomes.
Some didn't tried to expand, they just got well adjusted to their original territory and didn't need more resources.
But everyone in those groups eventually disappeared from history.
Even when the american continent was discovered, the remaining "true" mayas didn't remembered or indicated that those big monuments, cities ever existed already taken / hidden by the jungle. Many of those also, were just found literally hundreds of years after having seen the last true mayas.