Eovaldi started on Friday night against the Pirates, and he is still scheduled to pitch on Wednesday against the Cubs.
Players on paternity are given a minimum of one day and a maximum of three.
In the days leading up to the start, the team had been weighing fallback options in case Eovaldi's wife went into labor. They were not needed, as the right-hander faced the Pirates in a game Miami lost, 8-6, in 13 innings.
Before heading to the airport on Sunday, Eovaldi threw a bullpen session at Marlins Park.
Late Sunday morning, he headed to his home in Texas to be with his wife, Rebekah, who was scheduled to have labor induced later in the day.
"The plan for him is to make his next start," manager Mike Redmond said.
After the bullpen picked up five innings on Saturday, and 8 1/3 innings on Friday, the team is in need of a fresh arm.
Dyson, 26, pitched in five games for Miami last year. He is on the 40-man roster and offers versatility because he can pitch multiple innings, and even spot start.
Redmond joked about the timing of the roster moves, which brought in Dyson for a few days that Eovaldi wasn't scheduled to pitch.
"If we could just plan these guys' babies on when we need a fresh arm, we'll be in business," Redmond said. "It does give us a fresh arm down there when we do need it. Our bullpen has pitched a lot of innings. More importantly, we want him to enjoy the birth of his baby."
In 20 1/3 innings with New Orleans, Dyson has a 2.66 ERA with 15 strikeouts and seven walks. He last threw on June 9.
Previously pitching in the big leagues gave him a taste of what to expect.
"You get your feet wet and get some experience, and figure out what you have to do better," Dyson said.
Earlier in the season at New Orleans, Dyson was involved in an altercation with teammate Chris Hatcher, who also is with Miami now. Dyson sustained a broken jaw that required surgery on May 1.
"Everything is fine," Dyson said. "It was two months ago. It's definitely past us. We're just looking to help the team win and do what we have to do. That's the main goal."