I'd like to ask a question about peregrine falcons and what happens if they swallow the birdring of another bird they've caught.
Which community would be the right one for this question? It doesn't seem to fit in Biology.
While I have limited experience with Outdoors, I think it might be on-topic there.
Their [birds] tag states:
Egg laying, two legged, warm blooded, vertebrates with feathers. Use this tag to ask questions about our feathered friends. Questions on topic include birdwatching bush craft, hunting and any outdoors related bird based activity.
More interestingly, there is an apparently well received question about birds and tags: Where to report bird with leg ring id?. The answers there suggest resources to report the bird's ID; as you said you saw that happen on a live channel, you might know the nest's location, and it might be worth putting a notice to the scientists/rangers monitoring the birds (if anyone is).
I suggest asking something along the lines of:
I saw a bird swallow the birdring of another bird it was eating on live camera [YouTube link or something]. I am wondering whether that could harm them.
Can it actually cause harm or is it designed not to? (Researchers might have taken into account that the smaller birds could get eaten, when designing the tag)
In that case, or if I ever was to see such a thing myself while [hiking/hunting/whatever], should I report it, and how do I know who to contact?
I believe this can be simply addressed by "a pet is any animal kept in or around a household that is dependant upon that household for food, shelter, and care."
Note that I'm no regular of Pets.SE either...