That Mongolia vid is pretty cool. Powerful birds.
We have had them kill a few new calves. Some think eagles only eat fresh kills. They do prefer it but like already mentioned, they aren't that picky at times. I set a game cam near a horse we had die expecting to get some good coyote pics. Over the course of a month in the summer the meat was all eaten. I only had one coyote visit one night to eat out the ass a little. Every morning at daybreak like clockwork about 50 crows would show up and go to work on it. About 9am every day a couple huge golden eagles would arrive and scatter the crows. The eagles would really rip that horse open exposing new meat. They would eat their fill and leave. Crows would get back on it and all be gone by noon. Magpies would hit it for awhile. This went on till it was only bones and some hide.
I had a dump site for skinned carcasses on our property. A couple times while coming in with more skinned coyotes I'd scare off eagles that would take flight with a coyote carcass. Not sure how far they would get with it as it was in trees but they would carry it far enough to get out of sight and 50' in the air.