“Any Day Now” opens Friday at Sundance Cinemas; R, 1;37, three stars. I will host a post-show chat after the film’s 6:50 p.m. Tuesday screening.

“Any Day Now” has so many ways to break your heart that it seems less a question whether Travis Fine’s indie drama will do it, but how. This is a sensitively-acted film that engages directly with several “issues” that resonate on today’s op-ed pages — gay adoption, treatment of people with disabilities — but does so without being didactic or preachy.

The setting is California in the late 1970s, and Rudy (Alan Cumming) is a drag queen who dreams of a singing career, his fleabag apartment a marked contrast to his glamorous day job. Rudy meets a closeted assistant district attorney named Paul, who seems as uncomfortable as his skin as Rudy is at ease in his. They hook up, but are a little surprised to find that not only are they attracted to each other, they like each other, and a relationship starts.

One night, Rudy sees that his junkie neighbor is neglecting her son, a teenage boy with Down syndrome named Marco (Isaac Leyva). On a whim of goodwill, he takes Marco in for the night, and then when the mother disappears, takes Marco in for good. They move in with Paul, and the three become a family of outsiders. But in the late ’70s when homophobia is overt and institutionalized, the authorities would rather see a special-needs child in an institution than a loving home run by a gay couple.

Rudy and Paul face an uphill legal battle to keep Marco, and the movie keeps us guessing whether they will prevail or not. Writer-director Fine sets up a seemingly insurmountable set of obstacles, and almost everywhere Paul and Rudy turn, they face a cold, unfeeling bureaucrat. At times the film plays with our expectations about how legal dramas work; when the couple hires a flamboyant, crusading African-American attorney (Don Franklin), we think this is the moment when the tables will finally turn in their favor.

But “Any Day Now” isn’t that simple, or that immune to how a legal system that has prejudice embedded into itself operates. The film features deeply felt, lived-in performances from all three of its leads. This is really Cumming’s showcase, as he has to reveal several layers to Rudy — the tough-talking Queens cynic inside the drag queen, the caring maternal figure inside the cynic. It’s an extroverted performance — Cumming even sings several songs in the film, such as the one referred to i the title song.

And it matches up well with Dillahunt’s introverted performance. Dillahunt usually plays either goofballs (“Raising Hope”) or villains (you knew there was trouble coming the moment he showed up in “Looper”), and he’s very effective playing a closeted gay man who, if he can’t secure justice and equality for himself, went into the law to try and quietly secure it for others.

But the film’s secret weapon may be Leyva, an actor who does have Down syndrome, and plays Marco with authenticity and dignity. “Any Day Now” is an ode to human kindness, as well as an exasperated cry against a system seemingly designed to discourage such compassion.