How are excavation permits handled if the turn you dig in straddles a year-end?
I think the key phrase in the rulebook is this one:


  At the end of his turn, the player pays any costs in weeks he has
  incurred by moving his time marker.


By this token, you wouldn't be able to dig without permission just because, after the weeks taken up by movement, you would be "into next year".  If you start your turn in one year, then everything counts as taking place in that year; excavation permits can only flip at the end of a player's turn.  

If you have a permit, you can of course use it at the very end of one year, and then have it flip back up again at the end of the same turn...