The moves bring the team's roster to 51 players.
Mulligan had been signed to the 53-man roster on Tuesday, so this was a quick stay for him. He was considered a strong blocker who could help until Rob Gronkowski returns. The Patriots now have Gronkowski, Michael Hoomanawanui and Zach Sudfeld at tight end.
Cave, the former Notre Dame product, had been claimed on waivers from the Browns on Sunday. He could be a candidate to return to the team's practice squad if another team doesn't claim him and he has interest in staying in New England.
Also, the Patriots officially signed offensive tackle Jordan Devey to the practice squad after his original contract with the team was not approved.