Period Fic

A Period Fic usually takes the characters of a property and places them in a different time period than the setting of the story. It may involve time travel, but most of the time it doesn't. It's akin to a Period Piece, but combined with either Fish Out of Temporal Water or reusing characters as in the Star System.

May or may not include the Used Future.

While this is most commonly seen in Fan Works, it is most certainly not limited to them.

Not about averting No Periods, Period. Nor should it be confused with a Period Piece.

Examples of Period Fic include:

Comic Books

Gotham By Gaslight is a well-known "official" example, placing Batman (still Bruce Wayne) in a Victorian setting. Also had an animated adaptation.
Similarly, the Marvel 1602 miniseries by Neil Gaiman posits a timeline where the entire Marvel Universe kicked off more than three hundred and fifty years earlier than it did in the canonical 616 timeline.

Fan Works

Rendezvous With Fate by "iCe" relocates Ranma ½ to feudal Japan, and casts Akane as the time-traveling widow of Ryouga, who has accidentally come back in time and possessed the body of Ranma's estranged wife, Kodachi.
Golden Day by Gondolier moves the plot of Phantom of the Opera to Hollywood in the Roaring Twenties—with good period research, no less.
Starting a year or so ago^[when?], there was a sudden explosion of Cartman/Kyle South Park fics set in Nazi Germany, with Kyle in a concentration camp, hiding in Stan's family's attic, or otherwise persecuted, and Cartman as a sadistic Hitler Youth.
Scion: The Last Aeromancer is a story on Fanfic.net which turns Avatar: The Last Airbender into a Western fantasy while keeping the plot mostly intact. However, the author hasn't updated it since 2008, so it's sadly a Dead Fic.
NGE: A Century Apart answers the question, "What would happen if Evangelion took place 100 years earlier than it did in canon?"
Matrose Mond by David H. Olivier sets the action of Sailor Moon in Nazi Germany.

Film

The Phantom of the Paradise sets the story of The Phantom of the Opera in the 1970s Los Angeles Glam Rock scene (with a smattering of details from other stories, like Faust and The Picture of Dorian Gray, thrown in for extra flavor).