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From time to time, you have to start over (Like on Chester's update, you -have- to), but typically, your save files carry over without a fuss.  IF you try to delete the older versions though, make sure you find your save folder and copy it first.

Yeah I checked to see if it would carry over first and it didn's so I tried the folder method and it transferred, but it wouldn't let me progress through to the new update.