I have only shipped Cockatiels. I'm sure most any can be shipped with ease. When shipping across country, it is best to find the quickest way. Different airlines have different names for it. If they are shipped straight through with no change over, air freight is fine and it is cheaper.
Temperature is a factor.
It is best to ship when it is no hotter than 85 degrees and no cooler than 45 degrees. If the birds have to change planes, be sure you check the temperature at the point of transfer. Most airlines have a specific desk for live freight. You can call the airline number at your airport for directions. Be sure to tell the receiver to do the same at their airport and to be there by the time the shipment is scheduled to arrive.