We analyze the dynamics of Bose-Einstein condensates loaded in rotating ring lattices made of a few sites,and show how rotation maps the states found in this finite system onto those belonging to a static infinite lattice.Ring currents and soliton states in the absence of a lattice find their continuation in the presence of the latticeas nonlinear Bloch waves and soliton-like states connecting them. Both bright gap solitons and dark-solitontrains are shown to connect continuously to linear solutions. The existence of adiabatic paths upon varyingrotation frequency between states with quantized supercurrents suggests highly controllable methods for theexperimental generation of persistent currents