Whose descendants roamed the desert east of Palestine?

Though not all were nomads, a number of peoples lived there: the Edomites, Moabites, Ammonites, some of the Gibeonites (in Palmyra/Tadmor in later years), and probably others too. The region was called "the desert of nations" (Ezekiel 20:35) because of the (relatively) numerous peoples that lived there or traveled there. (Though this verse specifically refers to the Sinai, the desert east of Canaan/Israel/Palestine was nearby and shared the same properties.)

The Edomites were descendants of Esau (Genesis ch.36).

The Moabites were descendants of Lot (Genesis ch.19).

The Ammonites were descendants of Lot.

The Gibeonites were descendants of Canaan son of Ham.