growth
Once the first survivors of a crowd have gathered (nucleation), the crowd swells as more people join at the edges, and that swelling is growth. After a stable nucleus of the new phase exists, growth is the stage where atoms keep attaching to it, so the particle enlarges and the new phase eats into the old.
Most solid-state growth is diffusion-controlled: atoms must migrate to (or away from) the moving boundary for it to advance. When pearlite grows from austenite, for example, carbon has to diffuse so it can pile into the cementite lamellae and leave the ferrite lean. Growth rate therefore rises with temperature, because atoms diffuse faster when hot, which is the opposite of the nucleation rate's temperature trend.
The overall transformation rate is a tug-of-war between nucleation (favoured by low temperature and big undercooling) and growth (favoured by high temperature and fast diffusion). Because one wants it cold and the other wants it hot, the fastest overall transformation is at an intermediate temperature. This is exactly what produces the characteristic nose in a TTT diagram.
Just below 727 degrees C, pearlite nucleates slowly but grows fast, giving coarse pearlite; near 540 degrees C it nucleates fast but grows slowly, giving fine pearlite. Different lamellar spacings mean different hardness.
Nucleation and growth pull in opposite temperature directions.
Growth needs diffusion, so it slows as temperature drops; the martensite transformation is the exception, being diffusionless and growing at nearly the speed of sound.