Rizzi, 48, worked for three different head coaches during an impressive 10-year run in Miami. He interviewed for the Dolphins' head-coaching job last month before the organization decided to hire Brian Flores instead.
The Saints, meanwhile, decided to move on from their trio of special-teams coaches -- veteran Mike Westhoff, Bradford Banta and Kevin O'Dea -- despite New Orleans' improvement in many areas after coaxing Westhoff out of retirement as a midseason replacement in 2017.
Westhoff, who turned 71 last month, recently told NOLA.com that he had not decided yet whether he was going to come back in 2019. He now plans to retire and was "surprised and very disappointed" that the Saints decided to fire Banta and let O'Dea's contract expire.
Westhoff was considered a pioneer in the special-teams game during his long stints with the Dolphins and New York Jets from 1986 to 2012 before he retired the first time.
The Saints have one of the NFL's best punters, Thomas Morstead, and one of the best kickers, Wil Lutz -- and they led the league in net punting average in 2019 while allowing a total of just 60 punt-return yards. They also turned backup quarterback Taysom Hill into a special-teams star over the past two years, which included a blocked punt, a 47-yard kickoff return and three successful fake punts in 2018.
However, the Saints' return units have been just average, with none longer than 50 yards and no return touchdowns this past season. They finished on a low note by allowing a fake punt to turn the tide of their NFC Championship Game loss to the Los Angeles Rams.
It's also likely that the Saints wanted to pounce on a standout coach like Rizzi once he became available.
Rizzi began his tenure in Miami while current Saints assistant general manager Jeff Ireland was the Dolphins' GM. He also worked under current Saints assistant head coach/tight ends coach Dan Campbell when Campbell was Miami's interim head coach during the 2015 season. Rizzi also held the title of assistant head coach or associate head coach with the Dolphins over the past four years.
Among other highlights, the Dolphins led the NFL with 10 blocked punts over the past eight years. And they ranked fourth in 2018 in longtime NFL writer Rick Gosselin's annual special-teams rankings (the Saints ranked second).
Before joining the Dolphins' staff in 2009, Rizzi served as a head coach at the University of Rhode Island and the University of New Haven, among other college jobs.