Mii appearances

Mii is the players' avatar used in Nintendo Wii, 3DS, Wii U and Switch consoles. They first appeared on the Wii, where they could be created using the built-in Mii Channel and have since appeared in many games on all four consoles, as well as on the Nintendo DS. A Mii created by the player can freely be transferred by any of the four consoles to any other, as well as to the Nintendo DS.

Technically Miis are not characters, they're just avatars originating from the consoles themselves, so they don't have a series that's their own, and their appearance in other games can simply be considered that game using the console's avatar system. However it can be argued that once a player creates a specific Mii, it is a character, and it being able to appear in many different games can be considered a crossover of sorts.

Alternatively some consider Miis more or less as a race, so if a Mii appears in a game, while it's not a character from any other series, it shows that the Mii race exists in that game's universe.

If one chooses to consider either case, all games featuring Miis would be connected through an undirect link.

The games featuring Miis are (when not specified they are playable characters):

Wii games

Wii Sports (November 19, 2006) and all subsequend games in the Wii games series, except for Wii Chess.
Big Brain Academy: Wii Degree (April 26, 2007): Miis appear as students.
Mario Party 8 (May 29, 2007) and subsequent Mario games Mario & Sonic at the Olympic Games, Dr. Mario Online Rx, Mario Super Sluggers, Mario Kart Wii, Mario & Sonic at the Olympic Winter Games, Mario Sports Mix and Mario & Sonic at the London 2012 Olympic Games.
FIFA 08 (September 27, 2007) and its sequel FIFA 09.
MLB Power Pros (October 3, 2007) and its sequel MLB Power Pros 2008.
Rally-X Remix (October 23, 2007), part of Namco Museum Remix.
Pro Evolution Soccer 2008 (October 26, 2007) and its sequel Pro Evolution Soccer 2009: the player can choose to play with a team consisting entirely of Miis. Their bodies are designed much like regular players, though.
Smarty Pants (November 13, 2007)
We Love Golf! (December 13, 2007)
We Ski (January 31, 2008) and its sequel We Ski & Snowboard.
My Pokémon Ranch (March 25, 2008)
Magnetica (April 22, 2008)
TV Show King (May 12, 2008) and its sequels TV Show King Party and TV Show King 2.
Yakuman Wii (May 20, 2008)
Family Trainer (May 29, 2008) and its sequels.
Dance Dance Revolution Hottest Party 2 (September 16, 2008)
Samba de Amigo (September 23, 2008): Miis appear as background characters.
Bomberman Blast (September 25, 2008)
Tetris Party (October 14, 2008): Miis appear as the player's character beside the playfield, but obviously they're not controlled by the player.
Guitar Hero World Tour (October 26, 2008) and its sequels Band Hero and Guitar Hero 5.
Mario & Sonic at the Olympic Games (November 18, 2008) and subsequent Sonic games Mario & Sonic at the Olympic Winter Games, Sonic & Sega All-Stars Racing, Sonic Colors and Mario & Sonic at the London 2012 Olympic Games.
Pool Revolution: Cue Sports (November 18, 2008)
Pop-up Pirate (November 25, 2008): Miis can be used to replace the pirate.
Taiko no Tatsujin Wii (December 11, 2008) and its sequels Taiko no Tatsujin Wii: Do Don to 2 Daime, Taiko no Tatsujin Wii: Minna de Party 3 Daime, Taiko no Tatsujin Wii: Kettei-Ban, Taiko no Tatsujin Wii: Cho Gouka-Ban: Miis appear as dancers.
Let's Catch (December 16, 2008)
Deal or No Deal (January 13, 2009): Miis appear as the playable characters, but their bodies are redesigned to match the game's style.
Final Fantasy Crystal Chronicles: Echoes of Time (January 29, 2009)
Family & Friends Party (March 2, 2009)
Texas Hold 'em Tournament (March 27, 2009)
Texas Hold 'Em Poker (September 14, 2009)
NHL 2K10 (September 15, 2009) and its sequel NHL 2K11.
NHK Kouhaku Quiz Kassen (December 17, 2009)
And-Kensaku (April 29, 2010): while Miis appear mainly just as the player icon, a crowd of Mii is also shown in the background in the game's enviroment and therefore this counts as an actual appearance.
Disney Channel All Star Party (October 26, 2010)
Jeopardy! (November 2, 2010)
Wheel of Fortune (November 2, 2010)
Go Vacation (October 11, 2011)
Itadaki Street Wii/Fortune Street (December 1, 2011)
Soccer Up! (December 22, 2011)

Nintendo DS games

Personal Trainer: Walking (November 1, 2008)
Final Fantasy Crystal Chronicles: Echoes of Time (January 29, 2009)
Tomodachi Collection (June 18, 2009)

Nintendo 3DS games

StreetPass Mii Plaza (February 26, 2011)
Pilotwings Resort (March 25, 2011)
Pokémon Rumble Blast (August 11, 2011)
Tetris: Axis (October 2, 2011): Miis appear as the player's character on the bottom screen, while the playfield is on the upper screen, but obviously they're not controlled by the player.
Mario Kart 7 (December 1, 2011) and subsequent Mario games Mario Tennis Open, Mario Golf: World Tour and Mario & Sonic at the Rio 2016 Olympic Games.
Fun! Fun! Minigolf Touch! (December 13, 2011)
Sonic & Sega All-Stars Racing Transformed (February 8, 2013), as well as the subsequent Sonic game Mario & Sonic at the Rio 2016 Olympic Games.
Coaster Creator 3D (February 28, 2013): Miis appear riding the roller coasters.
Soccer Up 3D (April 11, 2013)
Tomodachi Life (April 18, 2013)
Disney Magical World (August 1, 2013)
Super Smash Bros. for Nintendo 3DS (September 13, 2014): Miis appear as customizeable characters, and as enemies wearing an exclusive outfit.
Nintendo Badge Arcade (December 17, 2014): Miis are seen walking around in the arcade.
Pokémon Rumble World (April 8, 2015)
Miitopia (December 8, 2016)

Additionally the character Nikki was created as a Mii mascot for the non-fictional Swapnote software released for Nintendo 3DS on December 22, 2011, and later received her own game.

Wii U games

Nintendo Land (November 18, 2012)
New Super Mario Bros. U (November 18, 2012) and subsequent Mario games Mario & Sonic at the Sochi 2014 Olympic Winter Games, Super Mario 3D World, Mario Kart 8 and Mario & Sonic at the Rio 2016 Olympic Games.
Sonic & All-Stars Racing Transformed (November 18, 2012) and subsequent Sonic games Mario & Sonic at the Sochi 2014 Olympic Winter Games and Mario & Sonic at the Rio 2016 Olympic Games.
Wii Karaoke U (December 8, 2012) and all subsequend games in the Wii games series (the only softwares not including Miis are Wii Street U and Wii U Panorama View, but they are not games)
F1 Race Stars: Powered Up Edition (January 16, 2014)
Super Smash Bros. for Wii U (November 21, 2014): Miis return from Super Smash Bros. for Nintendo 3DS with the same role.

Mobile games

The first mobile game by Nintendo is centered around Mii:

Miitomo (March 17, 2016)

Nintendo Switch games

Mario Kart 8 Deluxe (April 28, 2017)
Go Vacation (July 27, 2018)
Super Smash Bros. Ultimate (December 7, 2018)

Minor appearances

Miis appear as avatar icons in many games, but that doesn't count as an actual appearance.

Other appearances not as proper in-game characters are:

Wii:

WarioWare: Smooth Moves (December 2, 2006): Miis appear in certain microgames (which are fictional within Wario's universe).
Metroid Prime 3: Corruption (August 27, 2007): a bubblehead doll of the player's Mii wearing the Varia Suit can be unlocked in the game and will be placed inside of Samus' Gunship.
NBA Live 08 (October 2, 2007): a Mii appears as the coach superimposed onscreen.
NCAA Football 09 All-Play (July 15, 2008): a Mii appears as the referee superimposed onscreen.
Madden NFL 09 (August 12, 2008): Mii versions of the referee and John Madden appear superimposed onscreen.
Animal Crossing: City Folk (November 16, 2008): A Mii Mask is available for the player's character to wear. When worn it shows different emotions like a regular Mii, but it's still considered just a mask in-game.

3DS:

AR Games (February 26, 2011): the minigame "Mii Pics" lets the player take 3D pictures with an image of their Mii.
Animal Crossing: New Leaf (November 8, 2012): the Mii Mask returns from City Folk.
WarioWare Gold (July 27, 2018): one of the microgames (which are fictional within Wario's universe) is Mii Channel, based on the Mii-creating software and the player must choose the right facial features in order to create a specific Mii.