Today I move around 20 domain plus its content to new web hosting. Common process that should be accomplished :

On Old hosting account 

1. Backup database
2. Create archive of wordpress files + backup database
3. Scp to new web hosting through ssh

On new hosting account

1. Extract archive from scp
2. Create database, user access to database and add user to newly created database.
3. Import sql file (backup database) to new database.
4. Edit wp-config.php 
5. Edit ns record for domain and relax.

As I see on wp-config.php. This file not change from time to time. One of wp-config.php I see from old domain (registered on 2003) :

See that. No salt and no other option as I used to see on new install of wordpress. This file should be change to newest version. Simple way to do that is copy wp-config.php from newest installation and edit its content (specially credentials section). This word add more power and refresh the blog. I should add it to my to list from now on.