Has there been a movie star quite as adorable as Sam Rockwell? Granted, not every role he’s played lends itself to endearment — this is, after all, an actor who won an Oscar for his portrayal of a racist cop in “Three Billboards Outside Ebbing, Missouri” — but Mr. Rockwell invariably radiates a winning puckishness, an almost muppet-like malleability. 

Even as a voice-over artist, he convinces. The best thing about the animated series “F is for Family” is Mr. Rockwell’s turn as Vic Reynolds, full-time womanizer and blissed-out has-been.

In “See How They Run,” Mr. Rockwell is Inspector Stoppard, a Scotland Yard detective who has a suit just too big for his frame and a propensity for excessive tippling. A dab hand at comedy, Mr. Rockwell has some business here that is, in its dexterity and timing, positively Chaplinesque. Let’s just say that the fine art of the drunken pratfall is in capable hands.

Mr. Rockwell is at the center of an ensemble piece packed with actors who are game and, more important, adept at flexing comedic muscle. His co-star is Saoirse Ronan, the pensive young actress best known for the title role in “Lady Bird” and her portrayal of Jo in Greta Gerwig’s variation on “Little Women.” Ms. Ronan is Constable Stalker, an eager new recruit given to taking copious notes, jumping to conclusions, and possessed of an indomitable naivete.

“See How They Run” is an unceasingly clever contraption, a meta-parody of stage, screen, and literary tropes that nudge-nudges and wink-winks all the way up to and including the denouement. The proper use of the word “denouement” is, in fact, a running gag throughout the film, and pronounced with delicious overstatement by David Oyelowo, seen here as gadabout screenwriter Mervyn Cocker-Norris.

Screenwriter Mark Chappell and director Tom George have crafted what is, at first blush, an antiquated enterprise: an Agatha Christie whodunit. “See How They Run” begins and ends with the 1953 stage production of “The Mousetrap,” the West End theatrical production that went on to run a mind-boggling 27,000-some performances. 

A voiceover introduction, provided by Leo Köpernick (Adrien Brody), casts considerable shade on the play. He’s a Hollywood director, beloved by no one, who has been tasked to adapt the property for the silver screen. Except Köpernick is murdered, his tongue removed and the body deposited on the set of the play.

Before you can say “Sunset Boulevard” — you remember, the Billy Wilder film with its beyond-the-grave narration — “See How They Run” is off and sprinting with a host of allusions that even the most dutiful movie buff will have trouble keeping tabs on. Forget, for a moment, that one of the characters is Richard “Call Me Dickie” Attenborough (Harris Dickinson), and another is Agatha Christie (Shirley Henderson). The film also goes on to reference “10 Rillington Place,” both the film and the murders that served as its basis, and, in a stilted apercu, “The Shining.” 

“How would Agatha Christie like it,” asks a character toward the end of the picture, “if someone took one of her stories and twisted it and corrupted it just for the hell of it?” Messrs. Chappell and George indulge their fondness for the source material by giving it a 21st century twist that is, at times, indistinguishable from condescension.

As an exercise in promiscuous self-reflexiveness, “See How They Run” is expertly orchestrated and sometimes laugh-out-loud funny. Yet one can’t help but have a hunch that “Glass Onion: A Knives Out Mystery,” promised some time this year by Netflix, will be the real deal.