The lottery that takes place near the beginning of The President’s Cake is the kind you don’t want to win. That’s not because Shirley Jackson-style horror is brewing; the drawing in question is a matter of authoritarian fiat. The “prizes” are compulsory extracurricular assignments for grade-schoolers, decreed from on high to honor the nation’s leader on his upcoming birthday. For the resourceful 9-year-old at the center of this eloquent film, whose name is drawn despite her strategic prayers, this means baking a cake for her school’s mandated festivities.

The leader who has ordered the nationwide celebration of his birth is Iraq’s Saddam Hussein, and the setting is an unspecified point in the 1990s. It was a decade of several American-led military operations with “Desert” in their name, and a period when Iraqi citizens struggled under sanctions imposed by the UN — one of the endless examples through the ages of people being punished for their governments’ actions.

Hasan Hadi’s first feature, an Iraq-U.S.-Qatar production that includes well-known American filmmakers among its producers (Eric Roth, Chris Columbus, Marielle Heller), was filmed entirely in the writer-director’s native Iraq, its cast comprising mostly untrained actors. A stirring drama laced with humor, it’s a story of crossings: from home to school, village to city, one phase of life to another — passages emblemized in the canoes rowed through the calm waters of the country’s southern marshlands. That striking setting is first seen at dusk in a sequence whose fusion of poetry, tradition, kinship and omen sets the mood for what follows, as does the plangent lyricism of the (uncredited) score’s oud.

For Lamia (Banin Ahmad Nayef) and her grandmother Bibi (Waheed Thabet Khreibat), who live in a rustic village in the marshland, the girl’s luck of the draw is not just an inconvenience; it’s a hardship. The elderly Bibi has just been summarily dismissed from her farmwork in the fields (with a pitcher of milk as severance), and though the president has made water trucks available, such staples as flour and sugar will not be easy to find, let alone afford.

Most of the story involves the daylong city odysseys of Lamia, Bibi, Lamia’s classmate and neighbor Saeed (Sajad Mohamad Qasem), and Lamia’s gregarious rooster, Hindi (a sure contender should someone present an inaugural Palm Bird — or Palme Coq — at Cannes).

The day before the president’s birthday, the girl and her grandmother head to town with a shopping list of cake ingredients and Lamia’s book bag packed with an assortment of items to sell or barter. After crossing the water, they hitch a ride from an ebullient mailman, Jasim (Rahim AlHaj), who’ll cross paths with them again later, at a crucial juncture in the day’s events. Riding shotgun with Jasim is a groom (Thaer Salem) on the way to his wedding, even though he’s still healing from a life-changing injury caused by an American bomb, a situation that’s revealed with an impressive combination of gallows humor and spiritual equanimity.

Lamia is laser-focused on fulfilling her cake-baking duty, but Bibi has made this trip with another purpose, one that divides the two and sets Lamia on a quest through the city’s thoroughfares and backstreets and bustling souk, with its hawkers and their wares. Hadi stages the action here and throughout the movie with fluent energy, abetted by the agile camerawork of Tudor Vladimir Panduru (My Happy Family, Graduation).

With her book bag on her back and Hindi in a sling over her shoulder, Lamia sets out to find Saeed at the amusement park where he picks pockets while his father (Maytham Mreidi), who’s physically disabled, begs. Saeed is also one of the “lucky” ones, tasked with bringing fresh fruit to their school’s presidential birthday party. The odds, in his case, were stacked: He was forced to submit his name in the drawing five times, a punishment for tardiness meted out by their martinet of a teacher (Ahmad Qasem Saywan), the kind of man who declares, “I’m just a soldier and I must report anyone who refuses to obey” and who isn’t above stealing a kid’s lunch.

The fear their teacher has labored to instill in them has hit its mark with Lamia, who reminds Saeed that “the walls have ears.” She’s a wary rule follower who, in her increasingly desperate hunt for eggs, flour, sugar and baking powder, is challenged to embrace the idea that sometimes you need to break a few rules in order to comply with others. Saeed has a more irreverent attitude, and, in response to the old bromide “God is generous,” he utters one of the movie’s sharpest and funniest lines.

While Bibi raises hell in the police station, where her sense of urgency over a missing grandchild is met with snide indifference (“Now peasants think they can boss us around,” one officer snipes), the kids are on the go, indefatigable in their attempts to strike bargains with vendors. In the process, they tangle with the duplicitous, the conniving, the comical and the dangerous. There’s an earthly glow to grown-up decency and compassion when they make themselves felt — notably from Jasim and, late in the proceedings, Saeed’s mother (Nadia Rashak).

Lamia and Saeed are thoroughly engaging characters, and the young actors playing them convey strength and defiance beyond their years. Ahmad Nayef, in particular, navigates an intense range of emotions with no fuss, whether the scene calls for Lamia’s girlish delight when a café singer (Elaf Mohammed) draws her into a performance, or, later, her recognition that her life has irrevocably changed.

There’s lovely comedy in the way Lamia and Saeed often bicker like a long-married couple, and anguished pangs when their tensions explode in a memorable rooftop scene. But they’re also little kids, partial to staring contests while they navigate a world of absurdity. From the pastoral beauty of its opening sequence to the gut punch of its last, Hadi’s film is an exceptional screen debut, as perceptive as it is kinetic and, with one eye on the bombers overhead, brimming with life.