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I have used the IMAP migration tool to transfer people from Gmail in the past. There are some complications that revolve around the fact that messages often have multiple labels (including AllMail) and a generally weird IMAP architecture. Also, Google throttles big mail transfers, so the migration takes time with periods where things may seem to be stuck (Google not FastMail is responsible.) However, it is doable. I would not let the migration issues dissuade you if that is your only concern. Just recognize that it will take a some time.
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