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Abstract Anomaly-induced transport phenomena in presence of strong external electromagnetic fields are explored within a 4D field theory defined holographically as U(1) V × U(1) A Maxwell-Chern-Simons theory in Schwarzschild-AdS 5. Two complementary studies are reported. In the first one, we present results on the Ohmic conductivity, diffusion constant, chiral magnetic conductivity, and additional anomaly-induced transport coefficients as functions of external e/m fields. Next, gradient resummation in a constant background magnetic field is performed. All-order resummed constitutive relations are parameterised by four momenta-dependent transport coefficient functions (TCFs). A highlight of this part is a thorough study of non-dissipative chiral magnetic waves (CMW) in strong magnetic fields.