Prob leave alone if it’s wild and not in obvious distress. Have had poor luck with rehabing avian friends. They often seem to at first improve then I wake up to find they died in the night. Not always though.
Pigeons are often stunned or exhausted, especially racing pigeons. The last exhausted one we had in the garden was a racing pigeon, we chose to ‘save’ it from neighbour’s cat. Mrs R called the owner (from the number on tag) and he (in annoyed tone) said ‘don’t want her now’. He was quite happy to say that if we brought her back to him he’d ‘only wring her neck’.
We looked after her until she’s recovered strength then let her go and she flew into the tree, stayed an hour or so, and then flew off over towards the town. There’s a chance she unknowingly went ‘home’ to get her neck stretched, but (as the owner informed) also a chance not. Hard to know what to do in that situation but we gave her a chance at life at least.
Luckily our vet has a policy to assess injured wildlife and kindly euthanise free of charge. May be worth checking. That and local wildlife rescue.
If I’m in the field and find a very-injured bird with no way to get it to the vet then I will only kill it with a rock or branch if it is a small bird and the certainty of instant-kill. If it’s bigger than a sparrow then probably kindest and more-certain method is to dislocate it’s neck.
The easiest way with any bird up to the size of a mallard, is to grasp the body firmly in one hand, to stop it flapping, then take its head between the thumb and first two fingers of the other hand, and rotate until you feel the neck dislocate. Death is instant
What’s the most humane way to dispatch an injured bird?
Whatever you choose to do, I’m sure you’ll do it with kindness and the least amount of stress.