We show the existence of an exact mimicking network of $k^{O(\log k)}$ edges for minimum multicuts over a set of terminals in an undirected graph, where $k$ is the total capacity of the terminals, as well as a method for computing a mimicking network of quasipolynomial size in polynomial time. As a consequence of the latter, several problems are shown to have quasipolynomial kernels, including Edge Multiway Cut, Group Feedback Edge Set for an arbitrary group, and Edge Multicut parameterized by the solution and the number of cut requests. The result combines the matroid-based irrelevant edge approach used in the kernel for $s$-Multiway Cut with a recursive decomposition and sparsification of the graph along sparse cuts. This is the first progress on the kernelization of Multiway Cut problems since the kernel for $s$-Multiway Cut for constant value of $s$ (Kratsch and Wahlström, FOCS 2012).