One of my Laravel sites came over from a WordPress site with many users, so I created a custom UserProvider that extends the default one and only overrides the validateCredentials() method to check if the password saved for the user was hashed using WordPress so it can check if the password is correct using that algorithm instead.
The controllers in app/Http/Controller/Auth/ are there for you to modify. I wouldn't write something totally new but instead look at the traits they use and override anything you need.
What specific default Laravel behavior isn't compatible with the data you have?