Flying to Italy (from US) with small dog, will be obtaining pet passport there. Got everything lined up to do this but am having trouble finding some specific info about a connecting flight.

From what I can see the regs require getting to a vet in the first EU country you go to. Sometimes when travelling to Italy in the past we have had to clear customs in a connecting EU city such as ZHR. If I have a connection somewhere like ZHR am I going to be required to stop there and get a pet passport for my dog before continuing or will I be able to proceed to Italy? I am just concerned that once I go to travel from ZHR to FCO they will expecting different paperwork since I am no longer transiting and am now travelling from Switzerland to Italy.