A few options:
1) Arrange for the hotel in your destination city to hold the case for you and ship it there from your tour starting point.
2) Same as 1), but use a fold-up cardboard or plastic case to reduce the shipping charge.
3) Use disposable cardboard cases made out of bike shop boxes and a roll of duct tape (call ahead to have them save a couple boxes for you).
4) The Bike Friday method - a take-apart trailer frame that carries the case with your other luggage inside and is towed behind your bike.