How does food move through the intestines?

Peristalsis; it's forcibly moved through the digestive tract by muscle contractions.

After food is swallowed it is moved down the esophagus by peristalsis. The muscles in the stomach, small intestine, and large intestine continue the process. Food is further digested and broken down as it moves through the digestive tract, aided by digestive juices that are added along the way.