Radagast's ultimate fate

[Alcuin]

The purpose of the Istari was (UT, “The Istari”)

to advise and persuade Men and Elves to good, and to seek to unite in love and understanding all those whom Sauron, should he come again, would endeavor to dominate and corrupt.

…the memory of the Blessed Realm was to them a vision from afar off, for which (so long as they remained true to their mission) [the Istari] yearned exceedingly. …

…Radagast …. became enamored of the many beasts and birds that dwelt in Middle-earth, and forsook Elves and Men, and spent his days among the wild creatures. Thus he got his name … “tender of beasts”…

It wasn’t that Radagast was evil or malicious. Gandalf called him “the honest Radagast” in the Council of Elrond, and reiterated his trust in his integrity, pointing out that it was because of Radagast that Gwaihir the Windlord came to Orthanc, found Gandalf Saruman’s captive, and rescued him. Gandalf successfully used Radagast as a reference in his initial encounter with Beorn.

Radagast simply lost focus on his mission. He was sent to Middle-earth by Yavanna because Aulë sent Saruman. (This part of the legendarium smacks of the kind of jealousy seen in Greek myth between Zeus and Hera, for instance.)

I think that means that Radagast must have remained in Middle-earth until well into the Fourth Age. I don’t believe he still yearned exceedingly for the Blessed Realm, and being a Maia, he would not die of natural causes. The Men of the Vales of Anduin should have had some recourse to him for that time, as well as the Kings of Gondor and Rohan, if they thought to seek him out. Because of his relationship to Yavanna, one wonders why he was not an obvious ally of Treebeard and the Ents; but perhaps he was, and Tolkien did not inform us of it, or else it is not published. (There is a hint in Treebeard’s conversation with Merry and Pippin that he know might know more wizards than just Gandalf and Saruman, so it is quite possible that he knew Radagast, too.)

I think I met Radagast many years ago in Berkley, California. He was running a Ben & Jerry’s, wore Birkenstocks, had a staff, and was followed around by a couple of deer and lots of birds and furry critters.