While it's not easy to give an exact, single source for things like the bustles of the 1880s or the false rumps of the 1780s, that isn't one of them. Sticking to the 1880s bustle style, it somewhat evolved out of the rounder cage crinoline of the 1850s (something created to replace the layers of petticoats needed for that full shape). The cage gradually became more oval in shape through the 1860s, sticking out slightly more in back than on the sides. This change from emphasis on the hips to emphasis on the rear happens over and over throughout fashion history. In the late 1870s the natural form was in style, but the backs of the skirts were still layers of swags and drapes of extra fabric. The skirt drapes and styles needed more "canvas" space than the human body could provide eventually, so the bustle provided that addition. You can see the gradual evolution of the side silhouette through the century.

While there are definitely specific styles that are easy to point to as pulling from other cultures, I would't qualify this as one. Turkish influence brought about the pointed toe for shoes in the 1790s, Kimono inspired wrapping gowns for men in the 17th century, Turban fashions for women in the late 18th/early 19th centuries, or the Egyptian revival period of the 1920s are all good examples. In those cases it's possible to find a moment that brought about this fashion. The sudden rush in Egypt to find and open tombs, including King Tut's tomb brought about the Egyptian Revival style. The wrapping gown was introduce when Dutch traders were finally able to trade some items with Japan and brought the Kimonos back hoping to sell them for fabric, but finding people loved the style just as much.

It's interesting that this concept is still going around - I answered the same question on r/AskFashionHistory six months ago.