Final answer:
The pathway not involved in movement across capillary walls is active transport using proteins and ATP since capillary transport generally does not require energy in the form of ATP.
Step-by-step explanation:
Substances moving across capillary walls do so via various routes, such as direct diffusion through the plasma membrane, diffusion through the pores of fenestrated capillaries, transport via vesicles by endocytosis or exocytosis, and diffusion through intercellular clefts. These processes are facilitated by the unique structure of capillary walls, which is designed to allow the efficient exchange of substances between blood and tissues. Active transport using proteins and ATP, however, is not typically involved in this movement across capillary walls, as it refers to the movement of substances against their concentration gradient, which generally takes place in other types of cell membranes and requires cellular energy in the form of ATP. Capillaries facilitate the exchange primarily through mechanisms that do not require direct cellular energy expenditure.