It's not really much of a thermal differential though - that's the key. You need a thermal differential to generate energy. The more thermal differential there is, the easier it is to extract energy. A difference between the desert and average global temperature really isn't all that much. Turbines and the like prefer temperature differentials of hundreds of degree, with pressurized steam as the fluid of choice. It's relatively easy to vaporize and condense, it's really cheap to obtain, and after it's been cooled sufficiently, it can be dumped out into a natural sink, such as a large lake or river.
Air also doesn't have much in the way of heat capacity per unit volume. A better way might be to have pipelines of water, heated in the desert, pumped elsewhere.
But there would be significant problems:
1) The pipeline would need to be VERY heavily insulated with some exceptional materials. Aerogel comes to mind. Insulating a large pipeline would probably cost trillions of dollars. Aerogel is really damn expensive. The other option is to user "superinsulators," which are evacuated barriers with multiple sheets of material suspended in them to minimize heat loss by radiation. Also very expensive.
2) Pumping the fluid would take a lot of energy, likely more than could be extracted from it.
Using geothermal thermal maintenance systems would be more effective. Those are the things where you drill into the ground, down to where the temperature remains fairly constant, around maybe 60°F year-round. That helps cool a building in the winter, and heat it in the winter, reducing the demand on a different heat source.