This study is researching an experimental drug called dupilumab (called "study drug"). The study is focused on children with active eosinophilic esophagitis (EoE; an inflammatory disease of the esophagus) which impacts feeding and nourishment. The aim of the study is to see how safe, tolerable, and effective the study drug is when given for 24 weeks to children with active EoE. The study is looking at several other research questions, including: * What side effects may happen from taking the study drug * How much study drug is in the blood at different times * Whether the body makes antibodies against the study drug (which could make the drug less effective or could lead to side effects)