Greco-Roman mythology contains a story precisely about the region to which you refer. It so happens that this desertification narrative, which accounts for cold deserts as well as hot ones, is part of the Greco-Roman Flood myth.
Phaethon, the son of the Sun-Titan Helios and the Oceanid Klymene [Clymene], after being taunted by the accusation that he is a bastard, sets off to prove his paternity. After Helios swears to give him whatever he wants, Phaethon begs his father to let him drive his car, the fiery sun-chariot, across the sky for a day. Unable to convince him to make an alternative request, Helios hands the chariot over, and the universe is launched into cataclysm.
Unable to control the white-hot steeds pulling the chariot, Phaethon loses control of the vehicle, driving off the daily track and up too high into the sky. This creates the frozen regions at the edges of the world: the icy deserts. Angie Briese's Greek Gods and Goddesses website renders an interesting summary thereof:
Still, there are traces of Phaethon’s ride. The ends of the earth are
still covered with icecaps, and mountains still rumble, trying to spit
out the fire started in their bellies by the diving sun.
While the formation of cold deserts might be somewhat of a looser modern interpretation of the myth, the ancient poet Ovid quite undoubtedly refers to the origin of the Sahara in Book 2 of the Metamorphoses. In this sprawling epic, Phaethon also drives the sun-chariot too close to the Earth, and in so doing he reduces entire cities to dust and ashes, inadvertently annihilating whole nations after, essentially, setting the world on fire.
Every river and spring that doesn't burn now evaporates and the entire sea turns into a desert as well. The seven mouths of the Nile River "gaped dusty",
The wide seas shrink and, where ocean lay,
A wilderness of dry sand spread;
New peaks and ranges rise, long covered by the deep,
And multiply the scattered islands of the Cyclades.
The fishes dive, the dolphins dare not leap
Their curving course through the familiar air,
And lifeless seals float supine on the waves;
Even Nereus, fathoms down, in his dark caves,
With Doris and her daughters, felt the fire.
Poseidon himself huddles under whichever water remains because the sweltering air outside it burns with such unbearable intensity.
After the earth-goddess Gaia prays to Zeus for help against the raging fire, the king of the gods actually worsens the situation by zapping Phaethon out of the sky with a lightning-blast. This scatters the chariot, horses and charioteer all over the cosmos.
The chariot, crashing into Libya, ensures that the damage to the ground there is permanent, while the horses fall into the sea and Phaethon's corpse ends up in the Eridanos [Eridanus] River in Europe, where it smoulders for centuries thereafter, thus producing a set of pungent hot springs thereat.
To save the land from complete incineration, but also as an excuse to execute his side scheme to wipe out the human race, Zeus floods the Earth. The mass migration aspect of the story is implied only in this portion of the myth, as Deukalion [Deucalion] and Pyrrha are actually not the sole survivors of this segment of the worldwide disaster.
Those details, however, would not be pertinent to your question since the migrants are not from the newly-formed desert region, nor is their movement a result of the desertification aspect of the story; rather it is on account of the Flood.
Nonnus' Dionysiaka includes Indians in this desertification story, and Arabia features with some prominence in the same work. This implies that the sun-chariot-ride of doom is meant to account for desert regions all the way from Libya to Aithiopia [Ethiopia] (the latter of which is Phaethon's homeland) to Arabia and India (both Arabia and India often being considered to have been part of "Aithiopia" at different points in Greco-Roman thought).