We present results of the inversion of spatially averaged Stokes I and V profiles emerging from plage regions near disk center. The observations analyzed in this work were recorded with the Advanced Stokes Polarimeter. All atmospheric quantities determining the radiative transfer in the thin flux-tube approximation are inferred self-consistently with the help of the inversion code described by Bellot Rubio et al. With regard to thermodynamics, the retrieved model atmospheres are found to behave as expected on theoretical grounds. For the first time, velocities inside and outside the tubes have been derived empirically. The magnetic atmospheres resulting from the inversion are characterized by the absence of significant motions in high layers but show strong velocity gradients in deeper layers. These gradients turn out to be essential for reproducing the whole shape of the observed profiles and, in particular, the asymmetries and the extended red tail of Stokes V. Our scenario predicts that the Stokes V zero-crossing wavelengths of Fe I and Fe II lines are redshifted by small but nonnegligible amounts, which is indeed confirmed by observations made with the Fourier Transform Spectrometer. According to recent numerical simulations, the internal downflows derived from the inversion could be produced by the strong shear that takes place in the intermediate layer between the magnetized interior and the ambient medium. Another possible origin is magnetic flux undergoing convective collapse within the resolution element.