The Tsetlin library is a very well studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. In this paper we consider a generalization which can be interpreted as a self-organizing library in which the arrangements of books on each shelf are restricted to be linear extensions of a fixed poset. The moves on the books are given by the extended promotion operators of Ayyer, Klee, and Schilling while the shelves, bookcases, etc. evolve according to the move-to-back moves as in the the self-organizing library of Björner. We show that the eigenvalues of the transition matrix of this Markov chain are $\pm 1$ integer combinations of the transition probabilities if the posets that prescribe the restrictions on the book arrangements are rooted forests or more generally, if they consist of ordinal sums of a rooted forest and so called ladders. For some of the results we show that the monoids generated by the moves are either $\mathcal{R}$-trivial or, more generally, in $\textbf{DO(Ab)}$ and then we use the theory of left random walks on the minimal ideal of such monoids to find the eigenvalues. Moreover, in order to give a combinatorial description of the eigenvalues in the more general case, we relate the eigenvalues when the restrictions on the book arrangements change only by allowing for one additional transposition of two fixed books.