Transforming Polygons via Vertex-Centroid Moves

Let $P=(v_1,\ldots,v_n)$ be a polygon. In one move, choose $v_i$ and relocate it to any point on the line through $v_i$ and the centroid of $\{v_j:j\neq i\}$. Determine whether every initial $P$ can be transformed into a regular $n$-gon by a finite sequence of such moves.