Tropey Come Home

A boy and his pet are nearly inseparable, until the pet goes missing. For some reason, a character's pet will wander away from home, get kidnapped, or get himself lost some other way. This leads to two seperate storylines. On one side, the pet is incredibly lost, scared, and desperately trying to get home. The second story will involve the humans dealing with the loss of their pet, either by feeling miserable or by actively searching for them. In the end, the two parties will eventually reunite, often leading to a Crowning Moment of Heartwarming and maybe even An Aesop or two.

May overlap with I Will Wait for You where the human refuses to give up on waiting or to get a new pet. When the pet has been driven away by a cruel owner instead of lost, it's Come Back, My Pet. The human equivalent of this is There's No Place Like Home.

The Trope Namer is, of course, Tropey the Wonder Dog.

Examples of Tropey Come Home include:

Anime and Manga

Played with in episode 3 of Kamichu!: It's presented for most of the episode as a B plot to a god of poverty showing up, and while Yurie's plotline goes how you'd expect, her pet cat Tama's plotline has a twist to it.

Film

Homeward Bound: The Incredible Journey and its sequel.
The Adventures of Milo and Otis. Milo the cat gets lost, with Otis trailing him, and they are trying to return home.
Animal, Come Back Animal from the original The Muppet Movie fits here.

Literature

The original story Lassie Come-Home (1938) by Eric Knight was adapted into the beloved 1943 film Lassie Come Home, which is almost certainly the Trope Namer, and probably the Trope Codifier as well.
Ribsy by Beverly Cleary: Ribsy is left in the Huggins' car in the mall parking lot. He pushes the button that opens the window and gets out of the car, but when he tries to find his way back he gets into the wrong car and another family takes him home. He spends the rest of the book finding his way back home.
Pretty much the entire point of the short story Loveliness (1899) by Elizabeth Stuart Phelps. The titular dog is stolen and and his frail mistress, heartbroken, wastes away during the several months when he is missing. He gets found and everybody gets better and we learn that selling pets for medical experiments is bad .
There's a Jimmy Neutron storybook titled Goddard, Come Home! Goddard goes missing, but, with help from a beagle named Billy, Jimmy is able to track him down to the hideout of the tyrannical robots who dognapped him.
A somewhat odd example appears in James Herriot's books - a farmer sadly sends off his old cow to the meat market, and when she breaks away from the herd and runs home, he is overjoyed and vows that she will never leave home again. Besides being in the original books, it was slightly edited for a children's story titled "Blossom Come Home".

Live-Action TV

In the opening flashback of one episode of Pushing Daisies, Digby goes on a journey to find Ned at boarding school.
When Buck runs away, half the show is him in the pound, the other half is the family finally noticing he's gone, trying to replace him, arguing about going to find him, then finally picking him up at the pound.
The first episode of Skippy the Bush Kangaroo is about Skippy getting kidnapped and having to find his way home.

Newspaper Comics

At one point, Hobbes gets lost in the woods, and Calvin's parents have to go look for him at night. Calvin's mom even calls out "Hobbes! Where are you?!"

Video Games

The "Highway Star" portion of Elite Beat Agents revolves around a pug's efforts to return home after being accidentally driven hundreds of miles away.
If neglected, pets in The Sims 4 will frequently run away; sometimes they return bearing a gift-wrapped present of uncertain provenance.

Web Comics

When Chev "escapes" in Papi Nyang his owner spends a lot of time trying to find him, of course he's eventually trying to get home as well. Except she moved, to a place that would allow her to have cats (previously she was keeping her ownership of cats a secret).

Western Animation

Avatar: The Last Airbender; the episode 'Appa's Lost Days', where we follow the kidnapped Appa fighting abuse, starvation and loneliness as he tries to find Aang. For an episode about a character who can't even talk its incredibly moving.
Snoopy Come Home
Perry the Platypus in the Phineas and Ferb episode "Oh, there you are, Perry!". The song for that episode is actually called "Come Home Perry".
The Proud Family: Oscar accidentally loses Sugar Mama's poodle, who is naturally upset about this until his return.
A Pup Named Scooby-Doo: Scooby Doo runs away after feeling unappreciated, and of course, everyone goes looking for him.
SpongeBob SquarePants: "Where's Gary?"
Bolt
The Aristocats: Duchess and the kittens are kidnapped and misplaced by the butler, Edgar. They try to find their way home while their owner worried frantically about them.
- 101 Dalmatians also qualifies.
Done with a twist in The Brave Little Toaster, in which the "pets" are household appliances. Radio even broadcasts one such story, which serves as motivation for the others.
Spike leaves for an episode in Rugrats, for a long enough period for Stu to go through several new pets, including a tarantula and pair of gerbils. Subverted on the Spike half of the story, though: He gets taken in by a rich old man, who eventually brings him home. The man at the pound the Pickles go to even says this is a pretty common thing there.
Ace's A Day in the Limelight episode of Batman Beyond, where he goes after the guy who used him in dogfights, leaving Terry torn between tracking him down, and investigating the dogfighter himself.
Hey Arnold! episode "Abner come home".