Moore, 34, joins a Marlins team in the middle of a tight National League wild-card race. At 78-73, Miami is 4½ games behind Philadelphia, one game back of Arizona and trailing Chicago and Cincinnati by a half-game for the final slot.
Miami claimed Moore despite not being able to use him in the postseason. Players must be in an organization by Sept. 1 to be eligible for the playoffs.
Cleveland had claimed Moore, right-handed starter Lucas Giolito and right-handed reliever Reynaldo Lopez from the Los Angeles Angels on Aug. 31 in hopes of chasing down Minnesota for the American League Central title. The Guardians have stagnated and are seven games behind the Twins.
Moore, who will be a free agent this winter, is 4-1 with a 2.77 ERA and 57 strikeouts in 48⅔ innings. In five appearances with Cleveland, he allowed two runs in 4⅔ innings and struck out eight.