WordPress in separate directory

My usual workflow is create a development site on the back of my site e.g. mysite.com/dev/newsite and then when ready to go live it then goes to the new url e.g. newsite.com

However what I usually like to do is put WordPress in it’s own directory so it doesn’t clutter up the FTP e.g. newsite.com/site and then I add the index.php t root to access it from this folder.

Hopefully that makes sense! How would I go about moving the dev site to the new site using this structure?