Have the Dothraki visited Asabhad?

[Lord Varys]

If you reread AGoT you'll realize that the Dothraki must have had some success east of Bones and feel they can get that rather easily considering that Drogo originally intends to raid the lands along the Jade Sea before he decides to invade Westeros.

[Lordsteve666]

The way I've always seen it 1000's of years ago the Great Empire of the Dawn came crashing down, probably linked closely with or the cause of the Long Night. 

The fallout from that cataclysm forced most of the various people's living east of the Bones to flee west. Whatever mess was going down in the remains of the empire must have been bad, it pushed tribes and refugees to flee over the Bone Mountains, a treacherous migration as evidenced by the name given to mountains themselves. They literally died in their thousands to get away from whatever was going on. And we know it was bad because look at the state of Assai and the Shadowlands.  

From the world book we know the Dothraki remember their migration over those mountains. I personally think it was a sort of proto-Dothraki tribe that made the journey and the tales were passed down through generations until modern day Dothraki as we know them came to being. 

Once over the Bones the proto-Dothraki settled in the grasslands near Sarnor. For centuries they were the biggest threat to Sarnor and then once Valyria rose to power they were kept in check even more. 

Once the Doom happens they pretty much have free reign as the now dominant force in that part of Essos. So they destroy the Sarnori and move west again to dominate the void left by Valyria.