Yes, the Pokemon games turn off the 3DS WiFi. There are workarounds though.
So for X, Y and ORAS you need to turn off PSS in the PSS settings, save and then press the home button. Connect your NTR client, then re-enter the game. The connection should then stay active. I'm not sure about how to do it in Gen 7, let me look that up for a second
edit:
Unlike Generation 6 games, Pokémon Sun and Moon cannot be connected to the internet during normal gameplay, so after starting the application or performing any kind of reset you must connect PKMN-NTR at any of the following points: Title Screen. Continue / Mystery Gift / Live Competition Screen. Home Menu while the game is open and then wait until the top screen shows that you're connected to Internet.
For any of those games to keep working, you may need a memory patch to actually keep the connection active. Many NTR clients have those built in, though. Just try the above workarounds and if the connection still doesn't stay up, you may need to look into those patches