I know there's at least minimal info on Chult, Halruaa, and the Great Rift in the Grey Box. My copy is too buried currently to look it up , but I think Luiren, Dambrath, and Durpar also get short entries.
Other than appearing on the map, here ia the only thing the Grey Box says about Chult, feom page 5:
"Two of the maps are drawn to provide a general erview of the Realms, running from the Moonshae Isles in the west to the land of Thay in the east, and from the Spine of The World Mountains in the North to the Jungles of Chult in the south. These maps have been drawn with that information available to ourrepresentative in the Realms, Elminster the Sage, and represents what is known of those lands from the mindset of the Dalelands and Cormyr."
Halruaa merits two mentions, one on page 56 under Lantan:
"Finally, the people of Lantan have a mania for invention and devices. An Artificer, alchemist, or inventor who comes up with a new device will soon have a very persuasive Lantanna (or his agents) on his doorstep, making inquiries. It is known that Lantan gained knowledge of the printing press from the magical southern nation of Halruaa, and it is hinted that this aquisition was neither legal nor proper."
And on page 67 under the entry for Nimbral:
"The above may be all fairy-story, for many such tales come out of the south involving great and powerful magics that exceed the powers of the North. Yet just as often a flying ship appears on the horizon, carrying wonders from such a land as this or Halruaa, and so it is included herein."
The Great Rift ia on the map and merits the following mention on page 77, the third and final short paragraph in the e try for the Shaar:
"The region known as Eastern Shaar is sparser and less green, more of a wasteland. Within the Eastern Shaar is a huge cleft or canyon, known as the Great Rift, home to many southern dwarves."
Luiren and Durpar are mentioned on the map but not the text, and Dam rather has nothing.
So, all told, I really do expect the Beyond section of this book will have more on these places than the Grey Box did.