GIMP :: Smooth Edges Around A Cut-out

In GIMP 2.8, I've been having a great deal of trouble getting good edges around elements that must be overlaid onto new items.

Such items seem to need (a) a progressive blur around the edges and inward to its center, and (b) a decreasing opacity from the edges and inward toward the center.

I cannot figure out how to satisfactorily achieve this. I use this all the time, but now in 2.8 I cannot. (The only "solution" I can think of is using a fuzzy brush around the entire perimeter, but that does not blur it, and, of course, it takes forever....there's no point in even having a matte if you're going to do it manually!)