I raise pigeons. The dog training referred to is humane if done correctly, because the dog is not going to touch the pigeon. I know several dog owners who have a few homing pigeons of their own. A pigeon is put into a cage hidden on the ground under tall grass etc, and with a remote control door opener. The dog, a pointer type bird dog, is learning to smell out the bird and to go "on point" at it, that is, to freeze with the nose pointed at the bird telling the hunter where it is. On command, the dog is then to slowly move closer and "flush" the bird into the air for the hunter to shoot. In training with pigeons, the dog is told to "flush", and the trainer uses the remote control to open the cage. The homing pigeon flies into the air and back home to the trainer's loft where he has a nice supper waiting, and the dog is praised for correctly flushing the bird.
The inhumane hunters are the ones who go to a pigeon show or bird sale and buy whatever pigeon is cheap and use it this way. That pigeon is usually NOT a homer, and has no loft to fly home to, so after it is released it starves to death. In fairness, all of the guys I talked to at poultry auctions thought that all pigeons could could just find their way home. After I explained this to any number of hunters asking about buying my pigeons for training dogs, they have all stopped the practice and switched to using homers. A dog trainer really only needs a couple of birds, they don't usually take a bunch of dogs in the field at once, and it is a lot cheaper and easier for him to keep a couple in his garage or shed than to make repeated trips to bird sales and pay for many birds he can only use once. So there is no incentive for the hunter to continue once somebody explains this to him.
Sorry... maybe I was thinking about something else.
I have seen videos where people tie the pigeons legs together... and 'lock' the pigeons wings bending one behind the other... then thrown the live bird for the dog to retrieve..... Or they will use rubber bands on the birds wings to stop it flying a lot.
This is going to hurt the pigeon... and also it will be really stressful for it getting thrown.. and a dog grabbing it in its mouth.
What you describe sounds fine.