Rodriguez, 31, is dealing with left shoulder inflammation and his designation is retroactive to Thursday.
Rodriguez has no decisions and an 18.00 ERA in five relief appearances in his first season with the Red Sox.
Sherriff, 33, pitched in both games of Saturday's doubleheader split with the Tampa Bay Rays. He has no decisions and a 2.70 ERA in five relief appearances.
Ort, 31, is 1-0 with a 7.30 ERA in 12 relief appearances (12⅓ innings) with Boston this season.
Bernardino, 31, is 1-0 with a 2.93 ERA in 13 games (15⅓ innings) for the Red Sox in 2023.