Surface eigenmodes in a dusty plasma, where the dust is confined to a particular region, modelled as a uniform slab with non-uniform smooth boundary layers, are discussed. Inside this region, electron depletion occurs, while outside it, the plasma is just a normal uncontaminated plasma. In addition, the dust component can flow with respect to the background plasma, as a first approximation to cometary tails, where there is a notable difference between the fast flowing solar wind and the slow moving dust tail, viewed in the comet frame. The equilibrium densities and flow are non-uniform and the description involves a space-dependent dielectric function, which indicates the possibility of singular points, where eigenmodes resonate with local plasma oscillations. Surface eigenmodes on the slab are obtained analytically in the limit of long wavelengths, and these give rise to two distinct frequency domains, both having three different wave solutions. One of these refers to plasma surface waves, unaffected by the dust flow. The other two lead to convective surface modes, which become unstable for large flow speeds, akin to Buneman-type instabilities. A detailed study is made of the resonant processes in the boundary layers, including damping and growth rates.