A mongrel is where at least one parent is a cross breed. The word “mongrel” is objected to by some, but I see it as being much more accurate than “hybrid” that is sometimes misused. A hybrid is simply a cross between two pure races. I often hear the term “heavy hybridisation” by speakers, but what they really mean is “heavy mongrelisation”.
Owing to over a century and a half of importation, many of the bees in the United States are mongrels.
It is often said that it is hopeless trying to breed from or improve mongrels and that is probably true if you have a negative attitude. If you raise queens without any selection criteria you won’t get anywhere, but if you set out a reasonable list of characteristics you can make massive improvements in a short time.
It is not possible to backbreed, but some authoritive beekeepers have stated that if left alone for a considerable period any mongrels will eventually resemble the characteristics of the indigenous bees. Certainly that is my experience in my area,