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The high-entropy alloys Al$_{x}$CrFeCoNi exist over a broad range of Al concentrations ($0 < x < 2$). With increasing Al content their structure is changed from the fcc to bcc phase. We investigate the effect of such structural changes on transport properties including the residual resistivity and the anomalous Hall resistivity. We have performed a detailed comparison of the first-principles simulations with available experimental data. We show that the calculated residual resistivities for all studied alloy compositions are in a fair agreement with available experimental data as concerns both the resistivity values and concentration trends. We emphasize that a good agreement with experiment was obtained also for the anomalous Hall resistivity. We have completed study by estimation of the anisotropic magnetoresistance, spin-disorder resistivity, and Gilbert damping. The obtained results prove that the main scattering mechanism is due to the intrinsic chemical disorder whereas the effect of spin polarization on the residual resistivity is appreciably weaker.