The first teaser trailer includes same-sex couples as well as single parents of both genders as some of the possible "shapes and sizes" that parents and families can come in... and there's nothing wrong with that.
The movie proper tells how you are defined by yourself more than anything else. Another moral from the movie is not to judge others based entirely on what you hear.
Authority in Name Only: Lord Portley-Rind and his fellow White Hats. They appear to be the ruling body of Cheesebridge but are obsessed with cheese and their Nice Hats to the point that they neglect their duties to the city... and in Lord Portley-Rind's case at least, their families.
Curse Cut Short: Mr. Gristle lets out an "Oh sh—" before getting crushed to death by Snatcher's machine falling over.
Darkest Hour: Eggs and the last of the boxtrolls have all been rounded up, and it looks like all the boxtrolls have been killed, and Snatcher is about to kill off Eggs and get what he wants: a spot on the council and a Nice Hat. Then Winnie convinces Trout and Pickles to make a Heel-Face Turn, and the boxtrolls return for a Big Damn Heroes moment.
Disappeared Dad: Eggs' biological father is missing, presumed dead, after Snatcher and his men tried to take Eggs hostage. He turns out to be alive and well, still held hostage by Snatcher.
Disney Death: It seems that Snatcher was successful in crushing the boxtrolls to death, but it turns out they escaped in the middle of Eggs' speech, and they return in time to save the day.
Distant Reaction Shot: Snatcher's Death by Gluttony happens this way, with a cut to a distant shot of Cheesebridge and a huge cloud of orange-yellow mist erupting from the Portley-Rind mansion.
Drunk on Milk: Well, milk by-product at least; cheese makes Snatcher act like a surly drunk.
When Winnie takes Eggs to the party, he scratches his groin. She scolds him and tells him not to touch himself there in public: "That's why they're called 'privates'."
After Lord Portley-Rind realizes Snatcher is Madame Frou-Frou, he says "I regret so much."
Giant Food: The film features a giant cheese wheel called the Briehemoth. The White Hats bought it with money meant for building a children's hospital.
Go Mad from the Isolation: Mr. Trubshaw has gone more than a little dotty from being chained up and possibly tortured for ten years by Snatcher and his flunkies. He seems to get over it after Snatcher is defeated and he's reunited with his son Eggs.
G-Rated Drug: Archibald Snatcher is obsessed with cheese as a status symbol, even though eating it causes a violent allergic reaction that not only makes his face swell up in a nasty fashion, but causes him to act like a surly drunk and even hallucinate at one point. One of his lackeys, Mr. Pickles, even warns him "You know what cheese does to you..."
Have You Told Anyone Else?: Snatcher does a version of this. When Winnie discovers that Eggs is the missing child as the Exterminators are trying to catch him, Snatcher asks Winnie if her father is with her. When she says he isn't he tries to kill her too so as to blame that on the Boxtrolls as well.
Heel-Face Turn: Done by Mr. Pickles and Mr. Trout. Averted with Snatcher and Mr. Gristle.
Heroic BSOD: When Fish - the troll who raised Eggs for ten years; his adoptive father - is finally captured by Snatcher, Eggs seem to shut down for a little while before ultimately deciding to rescue him, thus finally taking action against Snatcher for the first time.
Humans Are the Real Monsters: Eggs briefly becomes convinced of this after his disastrous attempt to blend in with civilized people and Lord Portley-Rind's refusal to help.
Interspecies Adoption: Eggs is a Cheesebridge orphan raised since infancy by the eponymous Boxtrolls, to the point that for a long time he thinks he is one.
"You bit me, with your mouth" is first said by Winnie to Eggs, then by Archibald to Winnie.
When Winnie tries to get her dad's white hat, she is met by Archibald, who asks how the hat got all the way out on the street. Later, as Winnie tries to distract Archibald while the Boxtrolls dismantle his machine, she asks him the same question.
Jerkass Has a Point: During a pseudo-drunken rant, Snatcher gripes about how all Lord Portley-Rind and his fellow White Hats do is sit around and waste taxpayer's time and money on eating fancy cheeses, which is a startlingly accurate description.
Keep Away: Snatcher tries to take away Lord Portley-Rind's white hat when Winnie takes it and tosses it back and forth between the other good guys, as a distraction while the Boxtrolls dismantle Snatcher's machine from under him.
The Last Straw: Snatcher is already bloated from falling in the remains of the Briehemoth. Eating one more piece of cheese caused him to explode.
Leaning on the Fourth Wall: The ending has Mr. Pickles and Mr. Trout discussing how neatly everything has wrapped up, as if it was a story, "Assuming we're in one." The stinger goes on to have the pair debate their existence and whether all their actions are controlled by invisible giants. As the discussion continues, the audience sees the sped up movements of a stop-motion animator flitting about a miniature set, manipulating them both.
Left the Background Music On: When Winnie is directing Eggs to the Boxtroll Exterminators' lair on Curds Way, she points him towards a sign for Milk Street and tells him, "Milk turns into it." This is followed by a Rimshot caused by the one-man band tripping and dropping his cymbals.
Line-of-Sight Name: The Boxtrolls get their names from whatever words and images are depicted on their box, leading to names like Fish, Shoe, Fragile, and Knickers. The protagonist, a human boy raised by the boxtrolls, gets the name "Eggs".
Lovable Coward: The entire Boxtroll species. They’re completely unwilling to fight back against Snatcher. Eggs helps them grow out of it.
Missing Mom: Eggs' is never seen. Averted with Winnie, but Lady Portley-Rind isn't involved with the main plot.
Nice Hat: The members of the White Hat Council, the ruling body of Cheesebridge, wear very tall white hats as a status symbol; Snatcher and his lackeys wear similar red hats, and his obsession with moving up in the world includes getting a spot on the council and a hat of his own for wiping out the Boxtrolls.
Nightmare Fetishist: Winnifred is fascinated by all the gory tales of the boxtrolls and their rivers of blood.
Parental Substitute: Fish, for Eggs. Eggs indirectly calls the boxtroll his father at one point.
Plot Allergy: Archibald's severe lactose intolerance (he swells up from a tiny sliver of extra mild Cheddar) is part of what prevents him from joining the White Hats and eventually leads to his Karmic Death.
Scenery Censor: When the Boxtrolls come to Eggs' rescue, they don't have their boxes to cover them up anymore, so anything that isn't a butt is covered by scenery, props and other Boxtrolls.
Shut Up, Kirk!: Eggs tries to convince Snatcher that he doesn't need a fancy hat or fancy cheese to be somebody important, and that he is what makes himself. Snatcher retorts "I've already made myself, kid... this is my destiny!", and Death by Gluttony ensues.
Sommelier Speak: The citizens of Cheesebridge speak this way in regards to fine cheese.
Spider Tank: Snatcher builds one, with the help of captured boxtrolls, to break into the boxtroll lair and round up the rest.
Steam Punk: If the teaser and early artwork are anything to go by.
The Stinger: Mr. Trout and Mr. Pickles have an existential discussion, wondering if there really is free will and if they're just the puppets of some higher bodies. As they're having this discussion, the camera slowly zooms out to show the tiny set and the scene turns into a time-lapse of the animation of Trout and Pickles, still discussing the universe.
Stock Sound Effect: Baby Eggs' crying. In the flashback story, Baby Eggs was lowing to the ground in his Eggs box. The thunder booms, the lightning flashes, and rain pours down as Fish bolts through the gate and out of the yard with Baby Eggs still in the Eggs box.
That Poor Cat: The film features this among other sounds produced by a giant cheese wheel crashing through the town of Cheesebridge.
Unsettling Gender-Reveal: After Snatcher reveals himself to be Madam Frou-Frou, Lord Portley-Rind has a horrified look on his face and states "I regret so much".
Villain of Another Story: Lord Portley-Rind. Let's make a list. He wastes taxpayer's money and time just sitting around eating rare cheese, completely ignores his daughter to focus more on his hat and cheese, spends charity money for a children's hospital on a large wheel of cheese that most likely only he and his other White Hats would have eaten, completely ignores Eggs' public confession, tries to get out of his deal with Snatcher, when it is revealed the Boxtrolls are alive he flat out rubs the fact into Snatcher's face that he will never get what he wants, and even when his daughter was caught by Snatcher and he had to trade his hat to save her he tries to negotiate with him for something else. If Snatcher wasn't such a monster Lord Portley-Rind could have easily been the villain of this story.
Wrong Genre Savvy: Trout and Pickles, two of Snatcher's minions, are convinced they're the good guys fighting the forces of evil. Though both of them realize throughout the film that they are aware they are Wrong Genre Savvy.