The linear stability of natural convection in a tall vertical annulus with inner-cylinder motion was studied. The results show that the stability characteristics are significantly modified by the inner-cylinder motion. For a small Prandtl number fluid with a smaller radius ratio, the shear instability is dominant and the non-axisymmetric disturbance (m = 1) is most unstable no matter in which direction the inner cylinder is moving. For higher downward motion, the non-axisymmetric mode (m = 2) may replace modes m = 0 and 1 for some radius ratios. For higher Prandtl number fluids, the most unstable mechanism switches from a buoyant mode generated by the boundary layer near the outer cylinder to a buoyant mode induced by the boundary layer near the inner cylinder with an increase of curvature as the inner cylinder is moving.