Change of domain causes problems with linked media

Hi support-team,

we currently transformed our dev-environment into a live-environment. Therefore the backup of the dev-environment had to be implemented in the backend of the live-enviroment via All-in-One. After implementing the back-up into the live-environment, we’ve changed the URL of the new live and saved permalinks.

The domain of the new live-website was/is correct, but all linked pictures and sitelinks on the new live-website were not correct, that means that they still had the old URL.

We even tried to import the backup again (that means: URL of live-website was already changed), but the problem is still there.

We never had a problem by transforming a dev-environment into a live-environment before.
Where is the problem and how can you fix this without changing every link manually into the new URL (which we did now to solve this for a transitional period)?

Thanks!