Jeu de taquin is a well-known operation on standard Young tableaux that may be used to define an equivalence relation on tableaux of any fixed rectangular shape. Via the well-studied bijection between two-row standard Young tableaux and non-crossing matchings, jeu de taquin is known to correspond to rotation of the associated matching by one strand. In this paper, we adapt jeu de taquin to standard set-valued Young tableaux a generalization of standard Young tableaux where cells contain unordered sets of integers. Our modified jeu de taquin operation is shown to correspond to to rotation of various classes of non-crossing matchings by one strand. In the case corresponding to k-equal non-crossing matchings, closed formulas are derived for the number of jeu de taquin equivalence classes of standard set-valued Young tableaux.