After moving the Invision Community System to a new webhost (according to this document) some posts now contain random characters instead of the "original" characters.

For example: Â instead of a line break.
This does not occur when creating new posts.

How can I solve this problem?

You need to export the database and ensure you export with the correct charsets etc, then do the same when importing. You have missed some UTF8 data when importing, from what you are describing

The previous export was made using a windows sql client.
The problem was resolved by using WSL and the usual mysqldump/mysql utility.

Glad to hear you got the issue resolved 🙂