Anthony Van Dyck was a 17th century Flemish Baroque artist and Thomas Gainsborough was an English Rococo and Romantic artist of the 18th century.
How could the former have influenced the latter?

Van Dyck was born in Antwerp, and studied under Peter Paul Rubens. He traveled and worked in Italy before settling in England as the official court painter to Charles I (second Stuart King of England, king of Scotland, and king of Ireland).

British artist Thomas Gainsborough studied the portraiture of Van Dyck and although he painted King George III and his queen, he was not chosen as court painter. American born Benjamin West would make England his home and became court painter instead.

Notably, Benjamin West was one of the founders of London’s prestigious Royal Academy, along with Thomas Gainsborough.

Another similarity in the Van Dyck and Gainsborough story is that they were both knighted by their prospective kings; thus, Sir Anthony Van Dyck and Sir Thomas Gainsborough.

Van Dyck is best known for his relaxed elegance in portraiture and his flattering portrayal of King Charles I and his court. The supporters of Charles I were called "Cavaliers" as they were often of noble status and wore extravagant clothing.

A century later, Thomas Gainsborough would paint his famous "Blue Boy" (1770) wearing a fancy dress costume (in the style of Van Dyck’s Cavalier).
Note: the sitter's fancy blue bows on brown shoes.

Van Dyck would have another effect on the history of costume and style with his portraits of men with the now (world) famous short pointed beard that is called a "van dyck."

If you know of "Blue Boy" then you must have heard about its companion painting "Pinkie" by Thomas Lawrence at the Huntington Library in California. They have been displayed facing one another.

You can own a canvas giclee print of Thomas Gainsborough's "Blue Boy," available here from Amazon.com