The most common cause of swollen mesenteric lymph nodes in children is mesenteric lymphadenitis, which commonly occurs in children under 7 years of age after an upper respiratory tract infection or inflammation of the intestinal tract, where foreign bodies such as bacteria and viral products can spread to the mesenteric lymph nodes through the lymphatic vessels, causing the cells in the lymph nodes to proliferate rapidly and increase in size, resulting in swollen mesenteric lymph nodes.