We revisit the transport properties induced by the chiral anomaly in a charged plasma holographically dual to anomalous U(1)_V*U(1)_A Maxwell theory in Schwarzschild-AdS_5. Off-shell constitutive relations for vector and axial currents are derived using various approximations. a) In a weak external field approximation, the constitutive relations have all-order derivatives resummed into six momenta-dependent transport coefficient functions: the diffusion, the electric/magnetic conductivity, and three anomaly induced functions. The latter generalise the chiral magnetic and chiral separation effects. b) Nonlinear transport is studied assuming presence of constant background external fields. The chiral magnetic effect, including all order nonlinearity in magnetic field, is proven to be exact when the magnetic field is the only external field that is turned on. Non-linear corrections to the constitutive relations due to electric and axial external fields are computed. c) For a static spatially-inhomogeneous external magnetic field, gradient corrections to CME/CSE are analytically evaluated up to third order in the derivative expansion. Some of the third order gradient corrections lead to an anomaly-induced negative B^2-correction to the diffusion constant. We also find non-linear in B modifications to the chiral magnetic wave (CMW).