The statistics of randomly branching double-folded ring polymers are relevant to the secondary structure of RNA, the large-scale branching of plectonemic DNA (and thus bacterial chromosomes), the conformations of single-ring polymers migrating through an array of obstacles, as well as to the conformational statistics of eukaryotic chromosomes and melts of crumpled, nonconcatenated ring polymers. Double-folded rings fall into different dynamical universality classes depending on whether the random tree-like graphs underlying the double-folding are quenched or annealed, and whether the trees can undergo unhindered Brownian motion in their spatial embedding. Locally, one can distinguish (i) repton-like mass transport around a fixed tree, (ii) the spontaneous creation and deletion of side branches, and (iii) displacements of tree nodes, where complementary ring segments diffuse together in space. Here we employ dynamic Monte Carlo simulations of a suitable elastic lattice polymer model of double-folded, randomly branching ring polymers to explore the mesoscopic dynamics that emerge from different combinations of the above local modes in three different systems: ideal noninteracting rings, self-avoiding rings, and rings in the melt state. We observe the expected scaling regimes for ring slithering, the dynamics of double-folded rings in an array of obstacles, and Rouse-like tree dynamics as limiting cases. Notably, the monomer mean-square displacements of g1 ∼ t0.4 observed for crumpled rings with ν = 1/3 are similar to the subdiffusive regime observed in bacterial chromosomes. In our analysis, we focus on the question to which extent contributions of different local dynamical modes to the emergent dynamics are simply additive. In particular, we reveal a nontrivial acceleration of the dynamics of interacting rings, when all three types of local motion are present. In the melt case, the asymptotic ring center-of-mass diffusion is dominated by the contribution from coupling of the ring-in-an-array-ofobstacles dynamics with the tree dynamics. This contribution scales inversely with the ring weight and is compatible with a scenario in which constraint release restores a Rouse-like dynamics.