We had a major pigeon problem round our area.
A guy came and put up cages with food, the stupid birds just walk in over and over again.
The guy comes and takes them away. We haven't really asked what he does with them, but I suspect that they get their necks broken.
You can't just take them away and release them, the buggers either come back or become someone else's problem.
Your only practical solution is to kill them really, as quickly and painlessly as possible.