YES — with 2 exceptions:
YES — Your new country of employment cannot give you less credit for professional experience just because you gained it in another EU country. Comparable experience must be given the same credit when it comes to deciding salaries, grades and other working conditions. The same applies to access to employment in the public service.
NO — Any rule requiring you to live in your new country before being allowed to work in a particular public service would be illegal, unless justified by the specific needs of the position in question.