EU Pet Scheme
For a detailed explanation of the EU Pet Scheme, check https://ec.europa.eu/food/animals/pet-movement/eu-legislation/non-commercial-non-eu_en and/or https://www.gov.uk/take-pet-abroad. Be aware that if your pet is shipped as cargo over 5 days after your arrival in the EU, a commercial EU Pet Scheme form will be needed and the form will have to be issued within 48 hrs. of arrival to the EU. If your pet is only transiting the EU, this regulation will not apply.
Note regarding bloodwork:
The only approved laboratories in third countries are in Australia, Switzerland, Israel, and the United States. This means that those in other countries can expect to have to pay for the cost of the blood test in addition to the shipment cost.