Totally new here, I'll post an intro in a moment. But, to answer your question, it all depends. You're military right? If you are near a military exchange, then you should be able to get the usual (junk?) brands in the exchange or commissary. Local pet shops should also have some of the American brands (at least they did when I was in Germany). But, I never found a super quality brand either place. Our commissary in Germany did have Paul Newman's and Dick Van Patten (wasn't sure of their quality). But, I could get Science Diet, Eukanuba, Pedigree, Iams, etc. The problem with the local pet shops were the ingriedents were in the local language, which did me no good. I knew human food names in German but was lost looking at the side of a bag of dog food.
I did have a friend post on Facebook a few weeks ago that the exchange in Ramstein Germany had Blue Buffalo and that she had been ordering it from the US. If you have access to an APO box, you can try having it shipped to you. Don't know about the cost of shipping.
A word to the wise (I found out the hard way)--some countries are picky about bringing meat products in. As I recall, I had no trouble getting dog food into Germany back in 1998 when we first moved overseas with pets. When I returned to Germany in 2006, we didn't have any. But, when we returned in June to the US we had our beagle. I had food taped to the top of her crate (per the airline instructions) and they made me throw it out when we went through customs (it was considered a meat product) even though it was American dog food purchased in the PX.
You may want to see if they will move a few bags of food with your household goods, if you put them in airtight containers. Or, since you should have your mailbox before you go over, try having some food shipped before you leave.
Good luck on the move. I always wanted to be stationed in Italy, but never got anything but Germany!