In this article, a cell model is used for the evaluation of the alternating current (ac) mobility (dynamic mobility) of spherical particles in suspensions of arbitrary volume fractions of solids. The main subject is the consideration of the role of the electrical conductivity (SLC or K-sigma i) of the stagnant layer (SL) on the mobility. It is assumed that the total surface conductivity (K-sigma) resulting from both K-sigma i and the diffuse layer conductivity (K-sigma d), is constant in the cases considered and that it is the K-sigma i-K-sigma d balance that determines the SL effects. We first explore the effect of K-sigma i on the frequency dependence of the dynamic mobility. It is found that the mobility decreases on average, for any frequency, when K-sigma i increases. This is a consequence of stagnancy: ions in the SL, although contributing to the surface conductivity, do not drag liquid with them when they migrate and do not contribute to electro-osmotic flow or, equivalently, to electrophoresis. Three relaxations are observed in the mobility-frequency spectrum: inertial (the particle and liquid motions are hindered), Maxwell-Wagner-O'Konski (ions in the double layer cannot follow the field oscillations and can move only over a distance much smaller that the diffuse layer thickness l and the so-called alpha or concentration polarization process (the ions can rearrange around the particle, but they cannot form the electrolyte concentration field that appears at low frequency). Whereas the First two relaxations are little affected by K-sigma i, the alpha process undergoes significant changes. Thus, the mobility increases with frequency around the alpha relaxation region if K-sigma i is negligible, but it decreases with frequency in the same interval if K-sigma i is Finite. With the aim of explaining this behavior, we calculate the capillary osmosis velocity field that is the fluid flow provoked by the concentration gradient around the particle. The calculations presented demonstrate that the velocity is reduced (for each frequency and position) when the SLC is raised. It is proposed that such a decrease adds to that due to the changes in the induced dipole moment of the particle, also favoring a decrease in the mobility. These tendencies are also present when the volume fraction of solids, phi, is modified, although higher phi values somewhat hide the effect of K-sigma i, as in fact observed with all features of electrokinetics associated with the phenomenon of concentration polarization.