Dwarves never were the most populous race in ME, so they probably still survived even after the men became dominant. In fact, since there was no love between elves and dwarves, they were not allowed to flourish. When men took over, they would probably have more ways to progress their race.
As for the Eldar. They slowly diminished until all that would go had already gone into the West. Their were still elves in ME, maybe not many of the Noldor race, but at least some 'dark' elves. They would have known nothing of the ships going West. And we know that Arwen stays.
The hobbits stay in the Shire, and eventually become unnoticed... again.
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