For n = 3, a triangle can be made equilateral in two moves. Already for n = 4 the situation is less clear.
One could set many other transformational goals besides achieving regularity: scale the polygon by s > 0, rotate the polygon, etc. The notion generalizes to arbitrary dimensions.
A more difficult variant would be to use the area centroid rather than the vertex centroid, in which case m(t) does not remain on L, so that a vertex move would have the flavor of pursuit of a moving target.