"To flow or cause to spread every part of the body"?

Within a specific context, these words can all be used for this purpose:

"disperse"

2: to spread or distribute from a fixed or constant source: as

a archaic : disseminate

b : to subject (as light) to dispersion

c : to distribute (as fine particles) more or less evenly throughout a medium

"spread"

"saturate"

3 a : to fill completely with something that permeates or pervades b : to load to capacity

Definitions from M-W Online Dictionary

Permeate, in its sense “To enter and spread through; to pervade”, may serve. Also consider infuse, in its sense “To cause to become an element of something; to insert or fill”.

Perfuse, definition 3, means "Surgery. to pass (fluid) through blood vessels or the lymphatic system."

This contains the sense of something being spread through the body, as well as the sense of someone (the surgeon) causing it to flow.

I have heard it used on a documentary in the context of a surgeon reattaching a patient's severed hand, and the surgeon's sense of relief when he saw the blood perfusing the hand.

You might also look at this online medical dictionary for further use of perfusion.

The closest word might be Propagate which is used in biology often and means to spread and multiply. Another word that gets close is Saturate as in "saturate the body".