ESPN's Adam Schefter reported the news.
Wright, 26, blindsided Drake while his helmet was off Monday during practice. The team cut him a day later.
"It was time for us to move on. I don't really think I need to get into more detail than that," Dolphins coach Adam Gase told reporters.
Wright has played for the Detroit Lions, Cleveland Browns and Dolphins during his NFL career. He's played in 13 games, recording nine tackles and no sacks.
The Raiders placed Ahtyba Rubin on the reserve/injured list to make room on their roster for Wright. Rubin, 32, signed with the Raiders in June.
Oakland currently has eight defensive tackles on its roster.