We will present a simple setup in which one can make sense of a renormalization flow for FK-percolation models in terms of a simple Markov process on a state-sace of discrete weighted graphs. We will describe how to formulate the universality conjectures in this framework (in terms of stationary measures for this Markov process), and how to prove this statement in the very special case of the two-dimensional uniform spanning tree (building on existing results on this model). This is partly based on joint work with Stéphane Benoist and Laure Dumaz.