I asked Nintendo support if my Pokemon would be safe after doing a system transfer, and received this response:
"Thanks for writing. You will be pleased to know that your Pokémon will be safe during a system transfer, all you need to do to access them on your replacement Nintendo 3DS system is purchase a new subscription to Pokémon Bank."
Do I really need a new subscription to Pokemon Bank after a system transfer? Will my previous one no longer work?