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Neopets allows you to do this with their "Neopass" feature. You're able to connect your account through their mobile apps and transfer anything that you earn through their mobile games. Such as, Tales of Darcadia and Faerie Fragements.

Webkinz also has this feature as they have dual systems. They have a mobile app game and a browser based game. I think we'll see a cross promotion feature in the future where Neopets connects with Roblox or Fornite one day we'll they'll enable you to play as your Neopets in their virtual worlds.
 
Honestly I think web is already a flexible and cross-device platform. I plan on making layout for mobile for my own project and leave it at that. Tethering yourself to any distribution platform also means giving them a cut of whatever earnings from the project, and I'd rather keep that to the minimum and use it to keep the servers/me alive and not much else. That's just me personally though! Part of what draws me to virtual pet sites still is that they are indeed available on my main platforms without any hassle!