If you are going to shoot the birds, I'd just hand throw them. It's much faster and easier than fooling with the winger.
If you must use a winger...
pull feet back and fold behind the birds back
place bird sideways in pouch with head in direction the bird will fly on release
put in pouch belly down
rest barrel of gun (safety on) over top of bird until ready to go
If throwing shackled bird remotely to simulate a flyer, I may try to load the same way all thow I have no personal experience doing it. Biggest thing is to get the feet out from under the bird or they will kik and push their way free.
When I'm throwing roosters out of a winger I zip tie their legs together put their head under their wing and put them, I guess neck first, into the wingers. Feet sticking straight out. Loud noises will make them kick even tho they are "asleep" with the feet straight out there is a good chance they will stay in there. They do still times come out. Roosters are great for training birds!
I've had pretty good thrown birds with the Bumper boy pouch. Put the head under the wing, place bird in pouch head down at about a 45 degree angle, and strap them in. Be sure to release them as soon as you can, otherwise the birds get stiff and can't fly worth a darn....