Presumably they slowly get abandoned with fewer and fewer Elves and some or all of them becoming populated by men. The problem with a more certain and detailed answer is Tolkien writes little if anything about what happens after what is covered in Lord of the Rings, and even there, the appendix goes up to death of Aragorn and Arwen (and only very brief summaries of what happens over the rest of their life times). Other than Rivendell, which was always small and with a low population, these places would only reach that level of depopulation at some later point. Which leaves us guessing to some degree on exactly what happens and how. Technically Elven realms actually grow in this brief period covered (southern Mirkwood becoming 'East Lorien').
To address the point of Lorien mentioned in the comment there is a little bit in the Appendix, which may or may not agree with claim of it being empty by Arwen's death.
First from the end of the Tale of Aragorn and Arwen:
...Then she said farewell to Eldarion*, and to her daughters, and to all whom she had loved; and she went out from the city of Minas Tirith and passed away to the land of Lorien, and dwelt there alone under the fading trees until winter came. Galadriel had passed away and Celeborn also was gone, and the land was silent.
*Eldarion being the son of Aragorn and Arwen, and last king Arnor and Gondor to be named by Tolkien.
Then the note regarding the beginning of 'East Lorien' that I mentioned, from the Tale of Years.
In the North also there had been war and evil. The realm of Thanduil was invaded, and there was long battle under the trees and great ruin of fire; but in the end Thranduil had the victory. And on the day of the New Year of the Elves, Celeborn and Thanduil met in the midst of the forest; and the renamed Mirkwood Eryn Lasgalen, The Wood of Greenleaves. Thanduil took all the northern region as far as the mountains that rise in the forest for his real; and Celeborn took the southern wood below the Narrows, and named it East Lorien; but all the wide forest between was given to the Beornings and Woodmen. But after the passing of Galadriel in a few years Celeborn grew wary of his realm and went to Imladris to dwell with the sons of Elrond. In the Greenwood the Silvan Elves remained untroubled, but in Lorien there lingered sadly only a few of its former people, and there was no longer light or song in Caras Galadon.
Clearly the majority of the Elves of Lorien that had not sailed to the west had apparently moved to East Lorien. I am not so certain Lorien itself was completely empty when Arwen went there to die, but the population had shrunk drastically.
That is a little more about the fate of parts of the Elven realms but that is it, this is the last said of Thanduil's Woodland Realm and there is nothing to go on regarding Lindon, other than Cirdan intending to wait until the final ship left the Grey Havens to leave Middle-Earth.