Tableau switching is a well studied bijection on pairs of skew Young tableaux which swaps their relative positions. This is achieved by successively sliding the entries of the inner tableaux through the outer one via jeu de taquin (JDT) slides. Tableau coswitching is a similar but coplactic operation, meaning it commutes with any sequence of JDT slides. Coswitching is defined by first performing JDT rectification on the union of the tableaux, switching the resulting pair, then unrectifying the union. This definition requires us to perform large scale modifications to the skew shapes during the rectification and unrectification steps, which is both computationally taxing and obscures the effect of coswitching on the pair of skew tableaux. In previous work, Gillespie and Levinson define an algorithm which computes coswitching as a sequence of local moves, which do not alter the skew shapes of the tableaux, when one of the tableaux is a single box. In this paper, we extend the results of Gillespie and Levinson (and Gillespie, Levinson, and Purbhoo for type B) to the case the two skew tableaux are of arbitrary size. We also describe multiple bijections on tableaux, each descending to the evacuation shuffling (esh) operation on pairs of dual equivalence classes. We show how special cases of our local algorithm compute the Schützenberger involution on skew tableaux, as well as the wall-crossing and monodromy of certain covering spaces of the moduli space $\overline{M}_{0, r}(\mathbb{R})$.