Wormholes are theoretical products in general relativity, and are popular tools in science fictions. We know numerically the four-dimensional Ellis wormhole solution (the so- called Morris-Thorne's traversable wormhole) is unstable against an input of scalar-pulse from one side. We investigate this feature for higher-dimensional versions, both in n-dimensional general relativity and in Gauss-Bonnet gravity. We derived Ellis-type wormhole solution in n- dimensional general relativity, and found existence of unstable modes in its linear perturbation analysis. We also evolved it numerically in dual-null coordinate system, and confirmed its instability. The wormhole throat will change into black-hole horizon for the input of (relatively) positive energy, while it will change into inflationary expansion for (relatively) negative energy input. If we add Gauss-Bonnet terms (higher curvature correction terms in gravity), then wormhole tends to expand (or change to black-hole) if the coupling constant α is positive (negative).