Is their a way to move multiple vertices gradually at a time? I have moved multiple vertices gradually but a set of vertices at a time not all at a time, as shown in the animation below (First I moved the four vertices from top, then three, then two and at last the last vertex).
 I wanted to do the same thing as shown in the animation below but all at a time.
You can get the movement you want all at once with proportional editing.  


Select the top vertex
Press O to enable proportional editing
Change the Falloff type to be either linear or Smooth.
Press G to move the vertices
While still moving the vertices, use the mouse wheel to change the size of the influence until it covers all the vertices you want to move.