I read a travel article once about rescuing pufflings in Iceland. You go about collecting the ones wandering about town, put them in shoeboxes or something like that, then take them to the top of a cliff and hurl them off so they can fly out to sea. The woman who wrote the article said she didn't throw hard enough, and the puffling plopped to the ground and ran under the car, and her host's kids all laughed.
Apparently you really have to hurl them (underhand).
(Normally, they hatch in nests on cliff faces, then when they're ready they fly out to sea. But sometimes they land on land, because human lights can look like the moon on the water, so they get confused. And they can't launch themselves from sea level - they need to jump from a cliff to get airborne.)
In disasters, animal rescuers use Evac-sacs to hold cats and small dogs. They're like these mesh string bags in the article but are made of the mesh fabric that you use for sports jerseys. Pillow cases will work in an emergency, too.