The stratification with optical depth of the line-of-sight (LOS) velocity of a simple, isolated, round sunspot observed with the Advanced Stokes Polarimeter (ASP; Elmore et al.) presented here completes this series of papers that investigates the stratification in optical depths of such a typical sunspot. These results have been obtained through the use of the SIR technique (Stokes Inversion based on Response functions of Ruiz Cobo & del Toro Iniesta). From these data we have confirmed that there are strong downflowing velocities at log τ₅ = 0 that coincide spatially with the places where the magnetic field points downward (Westendorp Plaza et al.). Further confirmation is obtained by the application of the same method on a different sunspot, already analyzed with the Milne-Eddington inversion technique (Stanchfield, Thomas, & Lites). These downflows reconcile observations that have detected Evershed velocities outside sunspots together with suggestions of the possible return of the flow within the penumbra. The Evershed flow seems to be concentrated in elevated channels not thicker than 1 or 2 scale heights that are mostly located in the space between magnetic spines, i.e., in places where the magnetic field is more inclined, weaker in the inner-middle penumbra, and stronger in the outer penumbra and beyond the visible limits of the sunspot. This conclusion is based upon the tight correlation found between LOS velocities and the (reported in the second paper of this series) magnetic field strength and zenith angle. The upstreaming material is seen in the inner penumbra and the downstreaming in the outer penumbra. A strong increase with optical depth has been obtained for the LOS velocities that provides indications of the superposition of Evershed channels along the LOS. The differential opacity effect between the center-side and the limb-side penumbra, already reported in the second paper in this series, is also seen in the velocity maps and has suggested the comparison of the vertical mass flux through the upstreaming zones (mostly seen in the center side) and the downstreaming zones (mostly seen in the limb side), obtaining a fairly good balance between the two.