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Well, I can answer your question.  Each state has it's own regulations for moving and importing livestock.  While a good number of us consider our chickens and ducks pets or companions, they are still legally considered livestock like horses, cows, pigs and such.  If you show your birds, and go from state to state that  falls under different guidelines then if you are moving to a new state.  When you are showing chickens for example, you are not permanently moving to that state so "new" resident rules don't really apply since your there for a small amount of time.  However, horses are a little different in this rule as they have to be tested for and found free of certain diseases before they can enter that state for competition.
When you know that state you are moving to, then you need to contact that states agriculture and livestock department for the rules and requirements to permanently import your livestock to that state.  Chickens will have different regulations on vaccinations or tests then say ducks.  Each type of livestock and each animal you are importing must be tested and permanently identified.  The paper work must accompany the animals.  Either you have it (if your hauling) or your hauler will need the paper work.  It does not matter what states you are traveling through as far as what those states require for tests and such.  The only state the you need to comply with is one your moving to.
It is a good idea to make copies of all the paper work and tests in case on the paperwork getting lost or accidentally kept by an inspection point.  
The only other issue you have is where you are moving to in that state.  Will you need to find out if the house you will be moving to is zoned for chickens and ducks.   I also would highly recommend that you also find out the zoning or regulations of dog and cats at the same time.  Each county/city is different in the number of dogs and cats they allow in certain zones.  Some are really strict and some don't really care, but it is good to find out before you buy a house.
HTH