In the corridor of a St. Louis-area hotel last month, Sam Cash waited in line for autographs from members of the long-departed major league team known as the Browns. The signatures secured, he then sat at an adjacent table with his own Browns display that featured team trivia, a team time line and photographs of his favorite Browns player, Bud Thomas. Fans and several former Browns took a look.

Most of those attending the St. Louis Browns Historical Society and Fan Club’s annual luncheon that day were, understandably, in their 70s and 80s. Cash, a talented baseball player and Kansas City Royals fan, is 10.

Leaders of the Browns group — founded in 1984 on the 40th anniversary of the team’s first and only appearance in the World Series — know that Cash is an exception, that the clock is undoubtedly ticking on a 330-member organization dedicated to a sad-sack franchise that left town six decades ago to become the Baltimore Orioles....