The EU has issued instructions that people running from the war in Ukraine who have been granted temporary protection in one European country but then transferred to another European country, must obtain a new permit and document from them (the European country they decide to move to) as people under temporary protection. In these cases, the first temporary protection document issued should be destroyed.
In practice, this means that Ukrainians registered for temporary protection in Bulgaria could decide to move to another European country. There they will have the opportunity to re-register and enjoy the rights that the respective state gives to people under temporary protection.