Eagles have gotten progressively smaller and dumber over the years. Most of us just flew back to Valinor as the last threats to Arda disappeared. It's kinda sad to see your descendants devolve into wild animals, but thus is this life.
They probably flew to the Undying Lands sometime in the Fourth Age. That's just my theory, however.
Agreed. The Great Eagles either were allowed to fly to Aman or they eventually died out.
Maybe they eventually started shrinking in size gradually until they became normal eagles. However, if Middle-earth is our world in a bygone era, consider tales of the rocs, gigantic birds that could carry off elephants. It may be the giant eagles survived in some fashion.