What is nastic movement due to growth?
Think about say the left side of a shoot growing more than the right side. That way the left side will be longer and will be pulled back by the right side. This causes the shoot to turn to the right. By controlling the differences in the rates of growth in the two sides (using plant hormones) the plant can control exactly how much the shoot will turn. 

Most nastic movements however, are not directional. For example, in Thigmonasty, or the movement due to touch, of the Mimosa pudica (or the touch-me-not), cells at the base of the leaves pump out K+ ions and Cl- ions, while taking in Ca+2 ions, thus create a osmotic gradient and lose water. This loss in water results in a drop in turgor pressure of the leaves and they droop.

https://en.wikipedia.org/wiki/Thigmonasty#Thigmonasty_in_the_Fabaceae