The process involves extension and invagination of the plasma membrane to form small vesicles in pinocytosis (pinocytotic vesicles) or vacuoles in phagocytosis (food vacuoles).
Note the clear vacuoles in the colloid next to the epithelium where the increased activity of the epithelium to produce increased thyroid hormone has led to scalloping out of the colloid.
Vacuoles are often the largest organelle in a plant cell, and when plants do not get enough water, their vacuoles shrink, and the plant begins to wilt.