What are those tassle-like things that you see hanging down from near the tails of hawks and falcons? Are they part of the bird, or are they falconers’ straps?
I can’t say I have ever seen what you describe on a wild bird. I assume you must have seen one belonging to a falconer.
Straps. You never see them on wild birds, do you? 
They are used for restraining hunting birds during travel and before launching.
You can see the restraints in these photos from Scotland.
Actually I have seen them on a wild bird but I guess it must have escaped from captivity.
I have seen hawks and falcons carrying snakes, which could be mistaken for tassels at a distance. I have also seen other birds trailing fishing line they got tangled in, or carrying long strands of nesting material that might look like tassels. The OP is so vague it’s hard to know what he was seeing.
I don’t know of a raptor that sports any sort of tassle-like feathers beneath its tail.
A falconer’s escaped bird would fit the bill (the straps are known as jesses), but it should be noted that (contrary to what the OP seems to imply) these are extremely rare.
Yes, the OPs implication that this was something that had been seen regularly made me wonder if there might be some other explanation for the observations.
I came here expecting advice about hard drives.
You can see what I’m talking about in this picture.
I saw a large hawk in my backyard about 6 months ago who had those rope-like things attached. That was the only raptor I have seen up close in the wild. All the other ones were in movies or in live bird shows like at Busch Gardens Tampa.
I came here expecting information about velociraptors.
Those are definitely jesses, as would be the case in a picture posted on a falconry website. But, as previously noted, it would be unusual to see this in the wild.
There are something like 2000 falconers in the US, and at a generous guess 10% of them lose a bird each year (it’s probably a lot less than that). Most of these would be recaptured promptly. I do a lot of birdwatching, and go out with falconers perhaps 8 times a year - yet I’ve never seen a bird with jesses without its falconer nearby.
This is not by way of denying that you saw a falconry bird, but simply noting that such a sighting is far from routine.