Keeping the URL but changing the site – How?

Here you go, https://wordpress.org/plugins/all-in-one-wp-migration/
this plugin will come to rescue. I have migrated/transferred over 100s of WP websites with this superb plugin without having any problem. It easy, secure and helps in productivity. It will change all the urls which is necessary to be changed during migration, so after migration you don’t have to manually go into the code/db with change it. It does everything by himself.
Hope it helps.