Moore said the Royals attempted to complete a deal for Karns at last month's Winter Meetings.
"As is often the case, when two teams have mutual interest in a deal," Moore said, "you tend to stay on it."
Karns started the '16 season in the Mariners' rotation, then was moved to the bullpen. He finished the season on the disabled list because of a back ailment.
"Our medical people, [trainer] Nick Kenney and Dr. [Vincent] Key were comfortable with his health reports," Moore said. "He has had no restrictions with his offseason training."
Karns, who is under club control through 2020, had his best year with Tampa Bay in 2015, when he went 7-5 with a 3.67 ERA in 26 starts.
"I don't want to speak for [Mariners general manager] Jerry [Dipoto]," Moore said, "but I think giving up a player with that much club control was a hurdle."
Dyson, 32, became expendable when the Royals acquired outfielder Jorge Soler from the Cubs for Wade Davis at the Winter Meetings. The trade will ease the club's payroll -- Karns is in pre-arbitration and he will make somewhere near $550,000 this season, while Dyson is expected to make near $2.5 million in arbitration.
Dyson hit .278 last season with 30 stolen bases. Moore said it was difficult parting ways with Dyson, who was instrumental in the Royals' run of consecutive World Series appearances in 2014 and '15.
"He's been a great leader for us," Moore said. "He is valued very highly in the industry. His stolen-base ratio was always very high and he was a great defender. And last year, he showed he could get on base."
It's was a blessing from above to be apart of such a class act organization#KcRoyals I really enjoyed every moment I spent as a royal!!!! pic.twitter.com/H1K9zhGEFi