The pet will need a health certificate for the first country it will be traveling to and clearing customs in after it leaves the U.S. The requirements the pet will have to meet for travel to additional countries will depend on the mode of transportation (e.g. car, ship) as well as how long the pet will spend in each country.
It is recommended that you contact your local APHIS Veterinary Services Endorsement Offices for assistance with pet travel to multiple foreign countries.