When non-local species spread rapidly across large areas, what are they called?
If it is a non-local and rapidly spreading, then it is an invasive species.
The best example of this would be to look at Kauai, Hawaii. Only about one percent of the plant life there is native, which means the rest, although they have been there for a long time, are invasive species. For example the Albizia tree in Kauai is originally from Africa.
The people in Kauai at the time, I think they were the British, had cut down all the native trees for farming, but needed shade. They brought the Albizia tree because the tree has large branch spans that can cover large distances, without that big of a trunk. However, what they didn't realize was the difference in the intake of water Kauai offered, compared to Africa. Therefore, the Albizia tree spread rapidly and grew to enormous heights.
If I have gotten any of this information wrong, then I do apologize and hope that I did not offend anyone if I did. I would like to remind everyone that I am not an expert, but I do think I could share some knowledge, which is why I am here. Also, invasive species are usually hard to get rid of, because for one reason or another, they thrive.