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.
I think that David's solution would work very well for you. If you don't want to affect the nearest vertices just hide them with H. Changing the Proportional Editing type to Connected may be also helpful and work for you.