Region: Middle Earth
Adding a few quotes from LOTR just to illustrate what has already been written. The motivations for going to Valinor need to be looked at separately for elves, for maiar (Gandalf), and for hobbits (ringbearers).
I have always thought it interesting that elves use the term "exile" for Middle-Earth. Also Galadriel, even though she liked having her own realm away from the Valar, returns, once her ring has no power any more. Middle-Earth is a temporary exile. The voyage to Valinor is not going away, it's returning home.
Gildor: 'We are Exiles, and most of our kindred have long ago departed and we too are now only tarrying here a while, ere we return over the Great Sea....' FOTR book 1 chapter 3, Three is Company.
Narrator: Varda is the name of that Lady whom the Elves in these lands of exile name Elbereth. FOTR book 2 chapter 8, Farewell to Lórien.
Gandalf was sent to Middle-Earth for a task - as were the other, less successful or less faithful Istari. For Gandalf, even more so than for the elves, the journey is a return home.
Gandalf: '...I am going to have a long talk with Bombadil: such a talk as I have not had in all my time. He is a moss-gatherer, and I have been a stone doomed to rolling. But my rolling days are ending, and now we shall have much to say to one another.' ROTK book 2 chapter 7, Homeward Bound.
Does Radagast return to Valinor? Does anyone have an idea?
Frodo's wistful but realistic 'but not for me' illustrates it best. It's definitely an echo of Tolkien's war memories, I believe, as you have mentioned:
Frodo: But I have been too deeply hurt, Sam. I tried to save the Shire, and it has been saved, but not for me. It must often be so, Sam, when things are in danger: someone has to give them up, lose them, so that others may keep them. ROTK book 2 chapter 9, The Grey Havens.
I like how it's done in the ROTK film, when at the Grey Havens after a tearful goodbye, Frodo turns around once more with a blissful expression on his face. It gives the viewers hope that yes, even beyond that sorrow of parting, there is a prospect of contentment and bliss.
Then there's the lovely story of Legolas and Gimli going to Valinor together...