In the dermatology world, the link between diet and inflammatory skin disease (such as eczema and psoriasis) is still being navigated. There was a recent article on this published in the February 2020 edition of The Dermatologist.
A study by Dr. Aaron Drucker and colleagues showed little to no correlation between gluten consumption and the development of inflammatory skin conditions. This may not apply to individuals who are gluten-sensitive of gluten-allergic.
Eczema is closely linked with food allergies, and the role that food allergies play varies significantly from patient to patient. Food allergies can manifest immediately (minutes to hours) after ingestion or may take up to 48 hours to occur. Eggs, milk, wheat, nuts, seafood, and soy remain the front-runners in terms of foods that trigger allergic reactions.