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We construct the effective field theory for time-reversal symmetry-breaking multi-Weyl semimetals (MWSMs), composed of a single pair of Weyl nodes of (anti)monopole charge n, with n=1,2,3 in crystalline environment. From both the continuum and lattice models, we show that a MWSM with n>1 can be constructed by placing n flavors of linearly dispersing simple Weyl fermions (with n=1) in a bath of an SU(2) non-Abelian static background gauge field. Such an SU(2) field preserves certain crystalline symmetry (fourfold rotational or C_{4} in our construction), but breaks the Lorentz symmetry, resulting in nonlinear band spectra, namely, E∼(p_{x}^{2}+p_{y}^{2})^{n/2}, but E∼|p_{z}|, for example, where momenta p is measured from the Weyl nodes. Consequently, the effective field theory displays U(1)×SU(2) non-Abelian anomalies, yielding the anomalous Hall effect, its non-Abelian generalization, and various chiral conductivities. The anomalous violation of conservation laws is determined by the monopole charge n and a specific algebraic property of the SU(2) Lie group, which we further substantiate by numerically computing the regular and isospin densities from the lattice models of MWSMs. These predictions are also supported from a strongly coupled (holographic) description of MWSMs. Altogether our findings unify the field-theoretic descriptions of MWSMs of arbitrary monopole charge n (featuring n copies of the Fermi arc surface states), predict signatures of non-Abelian anomaly in table-top experiments, and pave the way to explore the structure of anomalies for multifold fermions, transforming under arbitrary half-integer or integer spin representations.