When someone - usually a male someone - decides to run for public office, one of the things he is expected to do during the campaign is kiss the babies of female potential voters. Somehow this will prove that he's not evil and can be trusted to run the [country/state/county/city/dog pound] properly.
Subversions in which the baby is shaken, autographed, or otherwise mishandled by a confused candidate are common
Related to Friend to All Children and Photo Op With The Dog.
A The Bash Street Kids strip has the teacher tell the class that they will be electing a head boy by democratic vote. Plug references the trope saying that politicians get votes by kissing babies, so he goes to kiss some at a nursery. But because Plug's ugly, the babies scream and gets thrown out without a kiss.
In Metroman's first appearance in Megamind. He juggles some babies, then kisses one of them in front of a huge crowd (who already loves him).
Film - Live-Action
In The Shaggy DA Wilby Daniels decides to run for District Attorney in order to clean up the corruption in the town. At one point he's talking with a female voter with a perambulator; the kid inside has a dirty, sticky face, but the woman gives him a look waiting for him to kiss her baby. He does so, leaving him with a facefull of chocolate. Just at that moment a photographer from the local paper decides to take Wilby's picture.
In the opening of The Emperor's New Groove, Kuzco is presented with three babies to kiss. He simply stamps them with a kiss mark stamper.
In The Hunt for Red October, NSA advisor Jeffrey Pelt tells Jack Ryan "I'm a politician, which means that I'm a cheat and a liar. And if I'm not kissing babies, I'm stealing their lollipops.".
In the 2012 film The Campaign the two candidates for office get into a fight over which is going to kiss a particular baby. Will Ferrell's character ends up punching the baby in the struggle.
In The Great Dictator, Adenoid Hinkel kisses some babies after a rally. The baby pisses on him.
In one science fiction short story, this caused an interstellar war. Upon First Contact with an alien race (that looked like teddy bears), an astronaut decided to do this. Unfortunately, humans looked like a kind of animal on the alien race's home planet that had the habit of carrying off babies to eat them and the astronaut's motives were misconstrued.
Bad Kitty for President makes use of this trope, as it's an election parody where Bad Kitty is running for president of the neighborhood cats. The narrator brings up the kissing babies trope and has kitty kiss the baby of the humans she lives with.
Live Action TV
The Late Show with David Letterman: In a joke about New Jersey Governor Chris Christie, Letterman said when running for office Christie "got confused. Kissed a blintz, ate a baby!"
Another Letterman gag: in "Top 10 signs [then Presidential candidate] Herman Cain is losing it."
Kissed a picture of himself and signed a baby
Parodied in an episode of It's Always Sunny in Philadelphia, where Mac tries to get a woman to give her baby to him so that Dennis can kiss it. Naturally, she's a little weirded out.
On Seinfeld Kramer runs for president of a retirement home's condo board; this involves treating some of the senior citizens there like the babies in this trope.
In the 1966 Batman series when the Penguin runs for Mayor of Gotham he kisses babies.
Mother: I gave him my baby to kiss, and he bit it - on the head!
In an episode of Sliders where Professor Arturo runs for Mayor of San Francisco, he screws this up by chasing after a woman shouting "Madam, I need to kiss your baby!"
My Hero has Piers Crispin kissing small children during his attempt to campaign for Mayor in one episode, though it backfires when he tries to kiss Ollie and gets punched in the face (Ollie being the son of George Sunday/Thermoman, and thus a superbaby).
Mentioned in a MAD Magazine article from 1962 (also an early version of the Russian Reversal): "Russian politics can best be understood by comparing them with American politics. For instance, in America, politicians have to kiss babies, and if they don't, the mothers can take their offices away from them. In Russia, the system is somewhat different. To get food, mothers have to kiss politicians and if they don't, the politicians can take their babies away from them."
Invoked by the cover art of Tankard's Two Faced album, which shows a literally two-faced politician holding a baby.
Pearls Before Swine made a parody of this. Rat was trying to win the election, but he couldn't come to a baseball event, so he sent Pig in his place to kiss a baby and throw a baseball out to the field. This cost Rat the election, because Pig kissed the baseball, and threw a baby out to the field!
An episode of Recess has the equivalent for kids; kissing pets. The boys tries to discredit Gretchen (who's running against Vince for class president on a platform of "Girls for Gretchen", although she genuinely wants to do a good job) by having her asked to kiss a pet lizard. She gets out of it by claiming to have a cold.
An episode of The Simpsons has Mr. Burns running for governor, and there's a scene where he mentions that he needs to go off and do this.
In an episode of Dog City, Dogfather Bugsy Vile rigs a mayoral election and wins with just five votes. Among his duties are kissing babies and christening a ship with champagne bottle. Hilarity ensues when he gets the two just a little mixed up....
Real Life (especially notable examples only)
Vladimir Putin was also known to kiss fish for this purpose.
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You are saying that you think this draft is ready to be published. That means the description is not ambiguous,
it doesn't duplicate an existing trope, there are at least three examples, and the title makes sense.
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