An up-down chain is a Markov chain in which each transition is a two-step process that moves up to a larger object and then back down to an object of the original size. The first goal of this paper is to present a general framework for analyzing these chains and computing their scaling limits. This approach unifies much of the existing literature while extending it in several directions. These include explicit conditions for constructing integrable up-down chains and convergence results for families of intertwined processes. The latter contribute to the method of intertwiners of Borodin and Olshanski. The second goal is to highlight a notable application of this framework to the settings of permutations and graphs. Here, we identify some integrable up-down chains and construct their scaling limits, a family of permuton- and graphon-valued Feller diffusions. Both the up-down chains and the limiting diffusions exhibit ergodicity, diagonalizable semigroups, and explicit expressions for the maximal separation distance to stationarity. For the diffusions, the stationary measures are the recursive separable permutons and recursive cographons recently introduced by the authors, and the separation distances turn out to be related to the Dedekind eta function.