What happens in the body after the microbes that produce food borne illness are swallowed?

Three choices. Three choices. First, the immune system generates a significant enouch immune response to kill them with their chemicals. If insufficient they can learn to live with us as a parasite feeding off of us. Or third they can actively go on to cause active invasion of tissues and organs.
Invasion. The pathogen can enter the bloodstream directly through the out pouchings called "villi" or it can break down the cellular tight junctions between the calls lining the gut and so allow for a "leaky gut" like syndrome. The latter can be a chronic source of inflammation and cause other illnesses.