Closest thing I can think of is the .hack games on PS2. You transfer your save through the 4 games in the original series. Then you can transfer your save to the first game of the GU series which just gives you some bonuses as well as giving the same name to a certain character as what you named them in the original games. Then the GU series has 2 more games after so technically the save transferred through 7 games.
For those wondering how the Pokemon transfers work:
Gen 3 -> Gen 4:
Need a DS with a GBA slot. Put the two games on the DS at the same time. There is a facility in Gen 4 that will let you send up Pokemon from the Gen 3 game.
Gen 4 -> Gen 5
Need 2 DSs (either can be a 3DS because of backwards compatibility). Gen 4 game on one and Gen 5 game on the other. There is a facility in Gen 5 for transferring. The games will connect wirelessly.
Gen 5 -> PokeTransporter -> Pokemon Bank
Two apps on 3DS. Gen 5 cannot interact directly with the Bank so you do the transferring on the Transporter app which sends the Pokemon to Bank.
Pokemon Bank <-> Gen 6/7
Gen 6 and 7 games can withdraw and deposit Pokemon into Bank. However, once a Pokemon has entered a Gen 7 game, Gen 6 can no longer withdraw it.
VC Gen 1/2 -> PokeTransporter -> Pokemon Bank
Similar process as Gen 5 except Pokemon from VC games cannot enter a Gen 6 game. The data is flagged as Gen 7 right away.
Pokemon Bank -> Pokemon Home
Pokemon Go -> Pokemon Home
Pokemon LGPE -> Pokemon Home
I guess for technical reasons they needed another service.
And finally, SwSh can withdraw and deposit Pokemon from Home.