Monadic Second Order interpretations

MSO can be used not only to accept/reject words, but also to transform words into other words, e.g. the doubling function w $\mapsto$ ww. The traditional model for this is called MSO transductions; the idea is that each position of the output word is interpreted in some position of the input word, and MSO is used to define the order on output positions and their labels. I will explain that an extension, where output positions are interpreted using $k$-tuples of input positions, is (a) is also well behaved; and (b) this is surprising.