The 23-year-old Bruun is the most expensive signing of the January transfer window in the Women's Super League.
Last season, Bruun won the French league with Paris-Saint-Germain before going to Lyon and now joining United on loan.
Bruun meets up with Caldwell and Moore, who joined the club on the WSL's Deadline Day.
After Lyon and United agreed on a contract, the Denmark international arrived in Manchester on Thursday morning.
"It's great to be here, I'm incredibly thrilled to be joining Manchester United," Bruun said on the club's website. It's a big club with a long history; I've been following the women's project for a while and seen how the squad has progressed, and I believe this transfer was a wonderful fit for me."
United had effectively confirmed the acquisition of defender Caldwell on a season-long loan agreement.
Caldwell, 33, can also play in the middle of the field. She had a great time in the United States with the North Carolina Courage in 2021.
Caldwell remarked, "Today is one of the most special days in my life thus far."
Moore, 31, was also signed on an 18-month deal from the Orlando Pride.
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