The Rays are taking on all of Duda’s salary, about $2.5 million, according to ESPN’s Buster Olney.
Expected to contend for an NL East title, the Mets started Thursday night 47-52, 13½ games behind first-place Washington and nine games out of a wild-card berth.
“I’m excited to join the Rays. They’re in the hunt,” Duda said. “Kind of mixed emotions. There are guys here I’ve grown pretty close to, and [the Mets] are a first-class organization. I was very proud to be a New York Met, and I’m gonna be very proud to be on the Tampa Bay Rays.”
Duda, 31, has spent his entire career with the Mets. He is eligible for free agency after the World Series, as are Mets outfielders Curtis Granderson and Jay Bruce, second baseman Neil Walker, shortstop Asdrubal Cabrera and relievers Addison Reed and Jerry Blevins.
Duda is hitting .246 with 17 homers and 37 RBIs.
Smith, a 23-year-old right-hander, was a third-round draft pick by Detroit two years ago and was dealt to Tampa Bay in April. He is 1-2 with a 1.60 ERA with seven saves in 31 games this season for four minor league teams ranging from Class A to Triple-A.
Earlier Thursday, the Rays acquired left-hander Dan Jennings from the Chicago White Sox for minor league first baseman Casey Gillaspie, the 20th overall pick in the 2014 amateur draft. Outfielder Shane Peterson was designated for assignment to clear space on the 25-man roster for Jennings and the 40-man roster for Duda.
The Associated Press contributed to this report.