Yes, in most countries you can relocate with dogs and cats, though the paperwork takes months of lead time, so start early. The standard requirements are a microchip, rabies vaccination given after the chip, and an official veterinary health certificate issued shortly before travel, often endorsed by a government veterinary authority in your departure country.
Some destinations, especially island nations and rabies-free countries, add a rabies antibody blood test with a waiting period that can stretch to several months, and a few still require quarantine. Certain dog breeds are banned or restricted in some countries, so check breed rules before booking anything.
Birds, reptiles, rabbits, and exotic pets are a different story entirely; many fall under wildlife trade rules and separate import permits, and some are simply prohibited. Airlines also have their own crate and weather rules. Get the official import checklist from the destination country's agriculture or veterinary authority, and consider a pet relocation agent for complicated routes.