Fister, 29, notched a career-best 14 wins with a 3.67 ERA in 32 starts and one relief appearance for the Tigers last season. The 6-foot-8 right-hander sports a 44-50 career record with a 3.53 ERA in 130 career games with Seattle and Detroit.
"Doug is battle-tested through playoff experiences, and the depth he brings to our staff is exceptional. We are thrilled to welcome him aboard," Nationals General Manager Mike Rizzo said.
Going to Detroit is Lombardozzi, 25, who averaged .259 with 22 RBI in 118 games last season, as well as a pair of 22-year-old left-handers, Ian Krol and Robbie Ray.