Schutzenberger extended his own operation of Jeu de Taquin on Young tableaux to linear extensions of arbitrary posets. We use this operation to define Markov chains on the set of linear extensions, and write down explicit formulas for their stationary distributions. When the posets are antichains, one obtains the well-known Tsetlin library. When the posets are the so-called down forests, we can prove formulas for the eigenvalues, with their multiplicities, of the generators. This is joint work with Steve Klee and Anne Schilling.