Your pet is part of your family, so you will be sharing the adventure of your expatriation in Switzerland altogether!
The household companion travels accompanied with its owner. If it is an unaccompanied entry (ex air freight), different rules and higher costs apply. Your moving company may take care of the relocation of the pet.
Depending on the kind, the age and the origin country, the process of importation varies as per the link here below: https://www.blv.admin.ch/blv/en/home/tiere/reisen-mit-heimtieren/online-hilfe-hunde-katzen-frettchen.html
Example: Coming from Great Britain, my dog is 1 year old and must have a Microchip. A anti-rabies vaccination is needed. The animal must be accompanied by a veterinary certificate and owner’s declaration. Pets entering Switzerland by air will be checked by customs (red, to declare). As a dog owner, you are required to register your animal in your Swiss place of residence. Your veterinarian must additionally register the dog in the Swiss dog database (AMICUS) within 10 days. Source and more details: BLV administration