1. There's no reason why they couldn't keep them! It would work the same way as pulling in possessions from their canon, since essentially the game becomes their canon in a game transplant as far as their present-day status goes.
2. It would depend on the power in question! A general rule of thumb, though, is that if it's a power tied specifically to the character (for example, a Persona for an import from Destiny Strings) they would keep it, whereas if it's a power that's entirely dependent on things within the setting to keep it going (for example, the sentient weapons from Exsilium that rely on the Initiative's technology) they would not. If you're unsure, explain as much as you know about the power within the application and the mods will be glad to let you know how it would translate into Rekindle!
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2. It would depend on the power in question! A general rule of thumb, though, is that if it's a power tied specifically to the character (for example, a Persona for an import from Destiny Strings) they would keep it, whereas if it's a power that's entirely dependent on things within the setting to keep it going (for example, the sentient weapons from Exsilium that rely on the Initiative's technology) they would not. If you're unsure, explain as much as you know about the power within the application and the mods will be glad to let you know how it would translate into Rekindle!