Its not a fine line, in my view. As long as its their chicken, on their lands, its their issue.
The moment it enters my lands, I will deal with their issue - and send them the bill, if I'm not feeling particularly neighborly.
If they abandon the animal on public lands? Well they no longer have claim to the bird, and have instead abused the Commons. I have no objection to exercising every legal remedy for that abuse, including refering the offender to the appropriate authorities. If they don't like it, they can petition to change the law - or convince a jury to nullify it in this particular case. I look forward to being called as witness.