[The Saucepan Man]

The descendants of Beorn, who dwelt in the Vales of Anduin between Mirkwood and the Great River, and who for many generations retained their ancestor's ability to take the shape of a bear.

I think that the Encyclopedia must be wrong on this. As Thalionmar originally asked, how could one man give rise to a group sufficient to merit its own title within 80 years? It does not follow from the references to "Beornings" in LotR that they are necessarily his descendants. In fact, when Aragron speaks of them in reference to the Rohirrim, he says:

... and their kinship is rather with the Bardings of Dale, and with the Beornings of the Wood, among whom may still be seen many men tall and fair, as a re the Riders of Rohan.

This speaks of a tribe far older than just three or four generations. No, I doubt that the Beornings were all descendants of Beorn, just as the Bardings would not all have been descendants of Bard. I suspect rather that they were a pre-existing folk who took their name from the one who rose to be their leader.

Whether some, or even any, of them shared Beorn's shape-changing ability is questionable. The paw-prints that Firnantoonion refers to could have been caused by normal bears, with whom Beorn would naturally have had an affinity. And Beorn's talent is marked out as special. I think that Tolkien refers to him as being a sorceror of sorts in one of his Letters.

So, I would guess that "Beornings" is the name used at the time of the War of the Ring to refer to the Woodmen who lived in the Anduin valley and western Mirkwood and that their name is attributable to the one who came to be their leader, and who perhaps united them in an identifiable grouping.