Saw a few eagles terrorizing migrating cranes in November. Couldn't figure out why 2 cranes were flying tight circles way up, seperated from their flock. 2 eagles had them scared $hitless. Finally they made a break and the eagles followed. I think the eagles were just showing their air superiority.
I have never seen it but I have read that a common tactic for birds to escape faster-flying falcons is to climb in a tight circle. Gulls eluding even peregrine falcons this way.
The raptors, being raptors, tend to have a heavier wing loading and many birds can exceed them in a spiraling rate of climb contest, presuming the chase begins where the falcon has no initial height advantage.
When peregrines and merlins do this to regular songbirds the songbird may tire or lose it’s nerve and attempt to dive back down with fatal results
