Durin - the Elves left Middle-earth largely because their part in the grand cosmic "scheme" of Arda (the "world", but more than just the planet) was finished. They were paving the way for Men, but they were destined to leave Middle-earth. The Valar themselves encouraged this, and eventually even those Elves who resisted realized that the time had come to give the world over to Men. Fortunately for them they still had a place they could dwell: the Undying Lands.
Durin wrote:
I have always been curious who would rule Middle Earth, whether it be Hobbits, Men, or Dwarves, or even Elves, who would come back in a New Age, with an un-explicable amount of Magic.
Hobbits - the Prologue to LOTR tells us that the numbers of hobbits have dwindled, and The Hobbit refers to their tendency to hide around the Big Folk. One would imagine this tendency has increased since there are no hobbits to be found these days (you may know of Tolkien's "conceit" that his writings were historical records of the long-distant pats of our own earth).
Dwarves - the draft of Appendix A.III given in
The History of Middle-earth XII states that eventually "the days of the Dwarves" would end. I take this to mean that they would die out, even though I don't particularly like this sad interpretation. On the other hand, this passage is not given in the published version, though one must still wonder where all the dwarves went. Perhaps they are still hiding belowgrounds.
Elves - before the end of the world (the extremely obscure Dagor Dagorath, similar to the Norse Ragnarok) I don't think this would happen. After that even though - the destruction of the world as it is known - what happens is anyone's guess. Before then though, the world remains under the Dominion of Men.