Hi Marian 😊,
Many thanks for your question. In France (not sure about other countries🤭) you need to carry your horse identification documents when travelling. There is no restriction regarding your horse's medical files except that compulsory vaccination have to be up-to-date (but this is general information and not really specific to transport).
However, if you transport a horse in order to compete, it is very important to make sure that all medical files are up-to-date as your horse will be in contact with other ones.
Hope this answers your question.
Have a nice day,
Anne-Sophie from Horsicar 🦄