I think the point of answer is hidden in your question. For diffusion transformation you need higher temperatures because diffusion processes are then much more easily. The non-diffusional transformations occurs at low temperatures because the diffusion is hindered.
In few words, at higher temperatures the distances between atoms increase (due to increasing of thermal vibrations of atoms) and atoms with higher energy and disrupted bonds can migrate over the structure. The higher the temperature the more atoms with disrupted bonds and consequetnly more significant diffusion does occur.