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Many interesting algebraic varieties appearing in low-dimensional topology and representation theory (for example various kinds of surface character varieties, or subvarieties of simple Lie groups or their flag manifolds) are known to admit cluster Poisson structures. Given some geometrically defined morphism between two such varieties, it is natural to ask whether it respects the corresponding cluster structures in a suitable sense. I will explain a kind of ‘gluing procedure’ for certain special kinds of cluster structures, which leads to a positive answer to the question above for morphisms of character varieties associated to cutting a surface along a simple closed curve, as well for morphisms between BFN Coulomb branches of quiver gauge theories obtained by restricting a factor of the gauge group to its maximal torus. Based on joint work with Alexander Shapiro.