The transfer to the small board
In the 14th century there were no draughts books, and in that time draughts was not described. But we know names for draughts from this time, and we can establish that in England as well as in France a new name for draughts was introduced.
The new name in England was checkers. The first reference we know is a poem from 1412. The poet took for granted that his readers knew what kind of game checkers was, and therefore we may say the game was generally known.
In France, the name for draughts became jeu de dames. The first reference was found in an English romance of chivalry from about 1380, where prominent knights play the game. This author too describes draughts as a generally know board game. It takes time before a game is generally known. That takes dozens of years, certainly in an age when it was difficult to travel and the communication was restricted.
For this reason it is allowed to assume that the English played draughts at the small board about 1350, and rather earlier than 1350 than after 1350.
Because the name jeu de dames is French, it is allowed to assume that also the Frenchmen played draughts at the small board in the middle of the 14th century.
By the way, dam(spel) is the Dutch form of the French game name (jeu de) dames. It was borrowed from French via Flanders. But Dutch players did not borrow the French variety, for draughts in The Netherlands and France was played with different rules. Holland played, for example, with a long king, France with a short king.