Kaleidoscopic groups are a class of permutation groups recently introduced by Duchesne, Monod, and Wesolek. Starting with a permutation group Γ, the kaleidoscopic construction produces another permutation group K(Γ) which acts on a Ważewski dendrite (a densely branching tree-like compact space). In this paper, we study how the topological dynamics of K(Γ) can be expressed in terms of the one of Γ, when the group Γ is transitive. By proving a Ramsey theorem for decorated rooted trees, we show that the universal minimal flow (UMF) of K(Γ) is metrizable iff Γ is oligomorphic and the UMF of Γ is metrizable. More generally, we give concrete calculations, in an appropriate model-theoretic framework, of the UMF of K(Γ) when the UMF of a point stabilizer Γc has a comeager orbit. Our results also give a large class of examples of non-metrizable UMFs with a comeager orbit. These results extend previous work of Kwiatkowska and Duchesne about the full homeomorphism groups.