Atlanta acquired right-hander Jackson Kowar in the Wright deal. Anderson was traded for $100,000.
A day earlier, Atlanta dealt pitcher Mike Soroka and infielder Nicky Lopez to the Chicago White Sox along with pitchers Jared Shuster and Riley Gowens and infielder Braden Shewmake for left-handed reliever Aaron Bummer.
The 28-year-old Wright is expected to miss the 2024 season after he had surgery in October to repair a torn capsule in his pitching shoulder. He was 1-3 with a 6.97 ERA in nine games with Atlanta this year.
Wright had a breakout performance in 2022, going 21-5 with a 3.19 ERA in 30 starts. He became the first Braves pitcher to lead the majors in wins since Tom Glavine in 2000.
Kowar was selected by Kansas City in the first round of the 2018 amateur draft. He went 2-0 with a 6.43 ERA in 23 big league games this year.
The 33-year-old Anderson was 4-0 with one save and a 3.06 ERA in 35 appearances for Atlanta this year. He missed the 2022 season following surgery on Oct. 27, 2021, to repair a torn right UCL.
Anderson is 11-6 with a 2.93 ERA in 128 relief appearances over four seasons for Miami (2019), Tampa Bay (2019 to 2021) and the Braves. He won Game 2 of the 2020 World Series against the Los Angeles Dodgers.
Left-hander Austin Cox was designated for assignment by the Royals to open a roster spot. The 26-year-old was 0-1 with a 4.54 ERA in three starts and 21 relief appearances for the Royals, making his big league debut on May 4.
Bummer has a $5.5 million salary next year in the final guaranteed season of a $16 million, five-year deal that includes team options at $7.25 million for 2025 and $7.5 million for 2026. Each option carries a $1.25 million buyout.
Kansas City also agreed to a $1.1 million, one-year contract with left-hander Josh Taylor and declined to offer contracts to right-hander Josh Staumont, left-hander Austin Cox, outfielder Diego Hernandez and catcher Logan Porter, allowing them to become free agents.
Atlanta declined contracts to right-handers Yonny Chirinos, Penn Murfee and Michael Tonkin, left-hander Angel Perdomo, catcher Chadwick Tromp and infielder Lucas Williams. Chirinos had been designated for assignment on Tuesday.