Spinoff Sendoff

On the pilot of a new Spin-Off, characters and settings from the original series will show up to give the new series a sendoff. Doing this has multiple advantages. For one, it provides the connection between the two shows making the second one canon to the first. For another, it increases the viewership from the original series as they tune in to see their favourite characters. In addition, doing so gives it some legitimacy in the eyes of the audience.

Compare Poorly-Disguised Pilot, where the new characters start off being introduced on the original series.

Film

 * For Beneath the Planet of the Apes Charlton Heston was only in it as a favor to the director. He wanted to die in the first reel, but they comprimised; he disappeared in the first reel and died in the last reel.
 * Star Trek gets one of these with Leonard Nimoy showing up to guide everything into place.

Literature

 * Several Star Trek novel series, just like the various TV series.
 * Captain Picard and the crew of the Enterprise-E feature heavily in the first Starfleet Corps of Engineers novella; Geordi La Forge hangs around for a few stories more, until the new characters are established enough to carry the series.
 * Picard and co are also heavily involved in the first Star Trek: New Frontier novel.
 * Kirk and company play this role in the first Star Trek: Vanguard novel.

Live Action TV

 * Each of the Star Trek spinoff series has one.
 * The Next Generation's first episode has a visit from The Original Series' Dr. Admiral McCoy, who inspects the Enterprise-D and gives it his blessing.
 * Deep Space Nine's pilot starts with The Enterprise-D docked with the eponymous station. Captain Picard appears and the new Commander Sisko promptly tells him how much he hates him. Sisko has some epiphanies, makes peace with Picard, and Picard gives him his blessing.
 * Voyager starts docked at Deep Space Nine, with Quark trying to con Harry Kim.
 * Enterprise is the only one not to be sent off by the previous series, but rather by one of The Next Generation's movies, First Contact, with a recorded message from Zefram Cochrane.
 * Stargate Atlantis starts with a visit to the arctic expedition by SG-1's Jack O'Neill and Daniel Jackson.
 * Stargate Universe - O'Neill picks up Eli, Jackson's on the info discs and Carter shows up commanding the Hammond.
 * Connor McLeod on the pilot of Highlander the Series.
 * It wasn't during the pilot of NCIS: Los Angeles, but somewhere early in Season 1 Abby from NCIS dropped in on the Los Angeles team.
 * Abby was also in an ad for NCIS LA before it first aired: "Are you guys following me?"
 * Director Vance also made several appearances in the first season. Which is understandable, as he IS the Director of all NCIS.
 * The Jeffersons: as Louise is packing the kitchen Edith Bunker comes over from next door to wish them luck in their new place.
 * In one of the pilots for The Muppet Show, Bert and Ernie from Sesame Street show up to perform a skit, where Bert worries that their little kid's show comedy wouldn't cut it on the new show meant for older audiences.
 * Prison-based Britcom Porridge was followed by Going Straight; in the first episode the protagonist Fletcher, freshly paroled from the original series' eponymous jail sentence, is escorted home on the train by prison warder McKay who had been a Porridge regular since the pilot episode.

Western Animation

 * Parodied and Lampshaded on The Simpsons on the "Spinoff Showcase" episode. For Wiggum's spinoff, Troy McClure hints some familiar faces might drop by to wish Wiggum luck. In the episode, The Simpsons appear in Mardi-Gras costumes and lampshade tropes going into the parody spinoff.
 * In same episode, the one with Abe Simpson as the matchmaking jukebox has Homer drop into the bar - at which point Abe furiously berates him for how outrageously cheap Homer was in providing for Abe's funeral. This ends when point Homer pulls Abe's plug out (again) and leaves.
 * The pilot episode of The Cleveland Show starts in Quahog as Cleveland tells his friends about his troubles and he decides to move away.
 * She-Ra began with He-Man journeying to Etheria to find Adora and give her the Sword of Protection allowing her to become She-Ra and do a Heel Face Turn going from a Force Captain of the Horde to leader of the Great Rebellion.