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With the transfer matrix method, analytical expressions of radially and azimuthally polarized chirped-Airy vortex beams (RPCAiVBs and APCAiVBs, respectively) through right-handed materials and left-handed materials are derived. We mainly discuss the effects of the radial and azimuthal polarizations as well as the effects of the chirped parameter on the propagation properties, including the propagation trajectory, the transverse intensity and the phase distributions, the peak intensity, the radiation forces, and the transverse orbital and spin currents of the RPCAiVBs and APCAiVBs in the x direction, the y direction, and their synthesizing direction. It is shown that the x, y, and synthesizing directions of the RPCAiVBs and APCAiVBs satisfy the vector synthesis rule, owing to the axial symmetry, and intensity modulation of the RPCAiVBs and APCAiVBs can be achieved by varying the chirped parameter.