Red vs. Blue begins with a comedic and zany plot for the majority of The Blood Gulch Chronicles. It then becomes almost completely serious during Out Of Mind and Recovery One. Finally, in Reconstruction, the drama meets the comedy in a batshit insane mash-up of genres.
This is one rare instance of the drama complementing the comedy. Wash's dead seriousness was entertaining in and of itself, and it also made Caboose's stupidity even more hilarious than it already was. The side stories similarly complement the main series; it is implied that the main characters are a source of comedy because they completely suck as soldiers.
Seasons 9 and 10 balance comedy and drama by splitting the time between the "past" of Project Freelancer (which we all know will turn out very, very dark) and the "present" of Blood Gulch in the capture unit (season 9) and the real Reds and Blues (season 10). The prequel stuff is a whole lot more serious than the present stuff, but there's still plenty of humor worked in, and there's still plot and drama in the present stuff.
There she is!! by SamBakZa started out as a silly romantic comedy about a rabbit-girl pursuing a cat-boy who finds himself falling in love despite his own prejudices and those of society. Then a rock crashes through his window at the end of the third installment, and the fourth sees the world go into all-out Fantastic Racism, with bad things happening to both the cat boy and the rabbit girl, and with things rather firmly in the Darkest Hour by the end. It all gets better at the end, though.
Nyan~ Neko Sugar Girls starts as a comedic series. Come episode 4, and it begins to get darker, with Raku-chan getting bitten by a squirrel and seemingly contracting rabies. The rest of the series features more Wangst until the last episode, which goes up and down in mood. Then comes the Sudden Downer Ending when Hitoshi has left Raku-chan for his kidnapper, which results in Raku-chan's Death by Despair.
Marvel/DC: After Hours had this in a big way. What started off as an uber-topical superhero satire slowly started to become a kind of uber-fanfic, placing the gamut of comic book characters in a world with very flexible rules. The first series only even begins to have a plot at episode three, the second series consists of five 20-minute episodes, and is so plot-centric that the jokes start to become slightly forced (most of them come from the Green Goblin being on tranquilizers, and then pretending to be on tranquilizers) it remains to be seen how long the creators can keep up the game before they run out of plot.
N.B.: they still do intermittent comedic side-series as well, which have thus far retained the comedic element completely.
It seems to be the method RandomGuy is adopting. Start off a new series with comedy and delve into darker elements by the finale, rise and repeat.
Chris Ushko's Ducktalez series got a massive dose of this. The original short was a crudely animated piece revolving around fart jokes, with the main story boiling down to Scrooge trying to kill Glomgold with a tank. While darker moments surfaced with Residuck Evil's horror imagery, Ducktalez 3 and The Duck Knight really saw this trope set in with Huey dying, Scrooge having and emotional breakdown and Quackerjack blowing up a gondola full of civilians. Vegeta acted very much as a typical Knight of Cerebus (though pretty much all his dialogue with Scrooge constitutes as a Crowning Moment of Funny.) Let's not even get started with the rather morbid scene where Huey finds all the costumes of the sidekicks Darkwing got killed over the years, or Quackerjacks' gruesome death.
The Frollo Show started out like any normal YouTube Poop. It then started to develop a plot, and became more action packed. By the time of the sixteenth episode of the series Frollo Gets Flashed By a Gothic Lolita, about 1/3 of the cast is killed off with two being irreversible.
The Ed Stories start out in blog format, then continue as a more formal type of prose fiction with a fairly whimsical tone (cf. "An Admin Password for the Universe"), then suddenly takes a turn for "the dreaded continuity", turns a hinted-at running gag into a major plot point for a longer story arc, and culminates in a Downer Ending.
Bonus Stage started as a funny, video game based, cartoon series, but took a turn towards serious right after Rya's death. The series was still basically a comedy after that, only much angstier and with more drama.
Oh, Doctor Horrible. The first act introduces the light-hearted tale of an incompetent supervillain, the girl of his dreams, and his cheezy superhero rival. Act Two starts with "My Eyes," Doctor Horrible's half of which at least is pretty dark, but really, it's just him bitching because Penny is going out with Captain Hammer instead of with him. The act then ends with "Brand New Day," which announces that Dr. Horrible intends to go through with Bad Horse's command: "There will be blood / It might be yours / So go kill someone! / (Signed, Bad Horse)" And then there's Act Three. Of course, considering the short length, it was obviously planned from the beginning
The Church Of Blow does this deliberately and with great effect; it starts off as a light satire of youtube vlogging, religion and cults, with episodes about deciding on the Church's logo (smiley face or weird mouse creature?). Then Cornelius Blow, the protagonist, dips further into insanity, the comedy gets darker and darker, someone shows up at Cornelius' house wearing his face, Cornelius kills at least two people before finally having a breakdown and discovering he's a fictional character and going off to find the real world. The whole series turns into an intelligent and elaborate parody and Take That of Youtube and everyone who uses it, raising questions about whether anyone's Youtube persona is actually the real them at all and if the very presence of a camera fictionalizes everything it records. Also it has lizard monsters, which may or may not be figments of Cornelius' imagination.
The Saga of Tuck has been accused of this, though the dark points of the plot have been implicit since day one. This didn't stop some fans from jumping ship.
Awkward. starts off as pure grossout humour but turns quite dramatic and serious as the series progresses.
Not even porn is immune to this. Summer Camp by Nick Scipio started out as an episodic, sex-laden Coming of Age Story about a boy being initiated into sex by his mother's best friend; but now, 4 volumes and a million words later, most readers are onboard primarily to find out who he marries and who died. (The interesting bit is that Nick planned it this way: the very first words of the story are a Framing Device in which both the wife and "Aunt D" are introduced by not named.)
Atop the Fourth Wall started out as an average geek reviewing bad comics on his futon and eventually made its way to said geek grappling with self-doubt, dethroning a multiversal conqueror, and commanding a massive starship. He still reviews bad comics on his futon, though.
Ah, the Anti-Cliché and Mary-Sue Elimination Society. Started up by three British girls with way too much time on their hands, with enough crack to make Scarface jealous. Now? It recently hit the two hundred story mark, with maybe two dozen writers, has an actual, slightly epic, plot, and (depending on the author) angst. Puh-lenty of angst. There's still a copious amount of crack, though.
Both New Prime and The Last Scene by Olan Rogers undergo this. The Last Scene started as just a nonsensical dialogue parodying action movie cliche`s in against a white background. Soon this white background became a plot point, and eventually it (almost) starts to take itself somewhat seriously. More so with New Prime, as it has now included plot twists, a (kind of) serious plot, with characters being Killed Off for Real. However, this trope is not entirely played straight as the series never lose their humor. New Prime takes itself more seriously than The Last Scene, as the latter moves more towards an Indecisive Parody than the original straight Affectionate Parody.
Not to mention what happens with the .MOV series, which started as a Black Comedy parody bordering Toilet Humour starring Applejack (Jappleack in the series). After a Kaiju version of Discord is unleashed in the world, it becomes an arc-driven story, Killing main characters, such as Apple Bloom and Rainbow Dash and transforming its universe into a Crapsack World.
The Web Video series Le Visiteur du Futur begins as a succession of one shot episodes where a time-travelling Cloudcuckoolander harasses the same everyday-guy, then quickly turns in a real plot, when antagonists (the Time Police) appears and the episodes lenght increases. It makes the series become a bit more serious, although it is still comedic.
Demo Reel got hit with this fast and on purpose. After the pilot - which focused half and half on movie parodies and brushing the surface of the characters - was released to mixed reception, Doug pulled the actual second episode and released the third in place of it. This one advanced the SWAG-as-Big Bad storyline and delved into the main characters' Dark And Troubled Pasts. The following episode, a bromantic version of Lost in Translation, was even heavier on the emotions and got mass praise.
For another Doug Walker series, there was The Nostalgia Critic, which started out with just the Critic being an asshole, and later got into heavy detail on the Critic's hatred of his job and himself. While still being funny, impressively.
Subverted in Treading Ground. After a number of opening strips starting as a sitcom, it seemed to turn into a big story about assassins. Then that turned out to be an unrelated side story and the main plot continued unabated.
Funny Business is written all as one chapter, note this is no longer the case now that it's been reuploaded but still undergoes this trope. The first part seems like one of those utterly forgettable children's books where the main character has magical powers. Once certain adults deduce the nature of the main character's powers, they interrogate her and she reveals that her cheery personality is fake and she has spent her whole life hating herself for something she did as a toddler, and which was easily fixed, to boot. The story of the main character's life then goes From Bad to Worse.
Your Favorite Martian started out entirely comedic, with song ranging from stalking your mother to getting buzzed on orphan tears. "Friend Zone" was what got them to be serious at first, being a song about unrequited love played straight, and they had very few straight-up comedic songs since, most either being Dramedy or complete drama, such as songs about being bullied in school ("Alien") and being in a borderline-emotionally abusive relationship ("Complicated").