Every country has different regulations regarding pet ownership and before you pack up your furry buddy, it's wise to research whether your new home is accommodating to animals.
In Asia, for example, pet-friendly accommodation can be difficult to find. In the UAE, dogs are banned from public areas and confined to designated dog parks. Some countries frown (or have an outright ban) on taking animals on public transportation.
Most countries require some sort of entry permit for pets. The application process varies substantially from country to country. If in doubt, then check whether your new country’s government website has a page detailing what you need to know about moving with your pet.