Netflix’s “In the Shadow of the Moon” gets started with a particularly grisly piece of business. One by one, an assortment of Philadelphians — a bus driver, a classical pianist, the guy grilling up a cheese steak at a greasy spoon — begin bleeding from their orifices before collapsing dead. At least the pianist is thoughtful enough to provide the paradoxically high-tone soundtrack for the carnage.

From there the film jumps through genres much as the narrative jumps through time. It’s a cop movie, with the cocky young Officer Locke ( Boyd Holbrook ) growing increasingly obsessed with what seems like an unsolvable case. It’s a serial-killer movie, although it never again approaches the blood letting of that opening sequence. It’s a family drama. Most surprising of all, it’s a time-travel movie, a sci-fi wrinkle that sneaks up on you amid the rest of the busyness.

Directed by Jim Mickle (“Cold in July”), “In the Shadow of the Moon” can’t quite confine its many moods within a single tone. The core, cat-and-mouse relationship, between Locke and the killer ( Cleopatra Coleman ), never really reaches the emotional depths for which it strives. Like most time-travel stories, “Moon” may prompt the urge to sketch and study a chronological plot diagram.