I used to live abroad, while I was on the register.
The local police I deal with in the UK had no problem with me moving. The only logistical thing to do was to do a foreign travel notification whenever I returned to the UK. If I didn't know the next time I would be visiting, they simply leave the return date blank with a comment.
I moved to a capital city and got a standard office job where English was the main language, but it obviously helps a lot to learn the local language.
With regard to checks, most EU countries do not have the austere level of checks that the UK has. In the country I was in (a western European country), employers are forbidden point blank from asking about criminal records unless you are applying for a job working with children, vulnerable people, security, etc. There was no "basic disclosure" or anything like that. Most EU countries are much more respectful of privacy.