What process is responsible for moving food along the esophagus and intestines?

Peristalsis

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"Peristalsis is the rhythmic contraction of smooth muscles to propel contents through the digestive tract. The word is derived from New Latin and comes from the Greek peristaltikos, peristaltic, from peristellein, "to wrap around," and stellein, "to place.""In much of the gastrointestinal tract, smooth muscles contract in sequence to produce a peristaltic wave which forces a ball of food (called a bolus while in the esophagus and gastrointestinal tract and chyme in the stomach) along the gastrointestinal tract. Peristaltic movement is initiated by circular smooth muscles contracting behind the chewed material to prevent it from moving back into the mouth, followed by a contraction of longitudinal smooth muscles which pushes the digested food forward."