Is there any form of cross-play supported between Pokémon GO and the handheld Pokémon RPGs such as X, Y, Omega Ruby, or Alpha Sapphire?
As far as I'm concerned, there are no plans for the future of cross-play, at least not something they announced.
There are, on the other hand, some declarations that were made from Nintendo that "they would like to add cross-play with Pokémon Sun & Pokémon Moon" but I assume they will decide to do it or not depending on the amount of people playing when the summer is over.
If I find the original interview where they said it, I will edit the answer to add some links.
EDIT: It seems that the team responsible for this project said in the E3 that they had the plans to tie-in both the new gen games and Pokémon GO:
"While explaining the uses of the GO+, the team also revealed that Pokemon GO and the GO+ will eventually tie-in to Pokemon Sun and Pokemon Moon. There was no explanation about how this will work at this point in time."
I still think that it will depend on the player base at the end of the summer, but we will see eventually.
There have been no announcements even hinting at this as a possibility, but other factors make this seem highly unlikely.
The underlying mechanics regarding stats and possible moves are completely different and would be difficult to translate from PoGo to the main series or vice versa.
Pokemon Go is made by a different company than the main series installments.
There is no interoperability with the core handheld RPG games, but it is possible to transfer your Kanto Pokémon from Pokémon GO to the Pokémon: Let's Go, Eevee/Pikachu games for the Nintendo Switch, where you can capture them in the GO Park. However, this is a permanent one-way transfer from your own linked Pokémon GO game to your Pokémon: Let's Go game. It is not possible to transfer in the other direction, or to trade/transfer to another player.
Here are the instructions from Nintendo's support website:
- Please note the following before transferring Pokémon from the Pokémon GO app:
- This feature becomes available when you reach Fuchsia City in Pokémon: Let’s Go, Pikachu! or Pokémon: Let’s Go, Eevee!.
- Pokémon originally discovered in the Kanto region (including their Alolan and Shiny variants) can be transferred from Pokémon GO. Special event Pokémon or Pokémon discovered in a different region cannot be transferred.
- Pokémon cannot be moved from Pokémon: Let’s Go, Pikachu! and Pokémon: Let’s Go, Eevee! to the Pokémon GO app.
- Pair your Pokémon GO account with Pokémon: Let’s Go, Pikachu! or Pokémon: Let’s Go, Eevee! on Nintendo Switch.
- Choose the Pokémon you want to transfer from Pokémon GO:
- On Pokémon Go, tap the Poké Ball icon, then select “Pokémon.”
- Tap the Nintendo Switch icon in the upper-right corner.
- Tap the Pokémon you want to transfer, then “Send to Nintendo Switch.”
- Select “Yes” when asked if you want to transfer the Pokémon to Nintendo Switch.
- Select “OK.”
- Set up your Nintendo Switch to receive the Pokémon:
- Launch your game on Nintendo Switch and go to the GO Park complex in Fuchsia City.
- Speak to the person at the counter.
- Select “Bring Pokémon,” then select a GO Park from the list.
- Select “Start Communication.”
- The smart device and Nintendo Switch will search for each other. When the connection is made, select “Yes” on Nintendo Switch to receive the Pokemon.
- Select “OK” on Pokémon GO to finish, and you can catch Pokémon you bring from the Pokémon GO app at GO Park.
This feature is going to be subsumed into the larger Pokémon HOME service coming in 2020, but it appears that it will still only be possible to transfer Pokémon out of Pokémon GO, not into it: