At the risk of sounding like a pessimist, I think you need to start by accepting that you will never be able to pasture your flock here in the desert the same way you can in lusher parts of the country. Throughout the year we do have native plants that grow well in accordance with seasonal precipitation, but they may not be sufficient to sustain even a small flock of birds. Right now the native mustard plants are growing strong and making my own chickens VERY happy, and the nearly indestructible Bermuda grass grows well with summer rains, again making for happy chickens. We also get things like wild plantain (though not as abundantly as other plants), wild amaranth, and a variety of desert wildflowers like lupine, calendula and poppies, but even with all of that the word "lush" never really applies here. I work VERY had to supply my birds with fresh grains, planting small pasture areas of clover, alfalfa, oat, wheat and a variety of sprouting greens (broccoli, radish, sunflower...) as well as planting things like kale for my birds during the cooler season before the heat sets in to make it bolt. I've had to grow most of the chicken greens in a separate garden area so I can cut it and ration it accordingly. Otherwise they'll completely destroy my entire garden in under three hours. (Yes, I know this for a fact.) And when we weed whack other parts of our property I always gather up the clippings and pile in up in various areas for the birds to enjoy.
I also grow heat loving plants like Armenian cucumbers in abundance during the scorching summers. I always get way more than my family and I can eat and toss the extras to the chickens, often chilled in the fridge first to help them combat the heat. Malabar spinach is another plant that once established grows exceedingly well in our desert heat in well composted soil and will provide a lot of tasty greens for both you and the birds. You'll need to trellis is somehow for best results though. The plant can become invasive on minimal water once established.
You also need to be keenly aware of the insect population out here. Fire ants are my number one complaint. My mature chickens pretty much avoid them with minimal complication, but those ants can quickly kill off chicks. Scorpions and centipedes may be a problem, but nearly all of my birds have adapted to see them as treats and consume them with relish. You may also have 'problems' with bees that hang out is substantial numbers around water containers. The chickens are typically smart enough to avoid them and I just place extra water containers to make up for the ones dominated by bees.
During wetter seasons you may be aghast at the number of roaches that appear. Don't fret! Chickens LOVE them, and I've trained mine to recognize that a low container left in place and then lifted up on cue will provide them with great fun and free protein as they chase those nasty little bugs around. It's pretty hilarious to watch. Two of my hens actually follow me around the yard to the areas where roaches are most likely to hide out in the daylight and wait for me to deliver their favorite treats. They're fearless little hunters!
You birds will also occasionally find small snakes, rodents, frogs and lizards out here, which makes for great entertainment for you and good nutrition for them. Watching them play "keep-away" while simultaneously bragging about what they have is downright hilarious, though watching their treats get pulled to shreds can be a bit gruesome.
Anyway, welcome and good luck!