can't login after move site to local - empty response
Finally i got it.
It was an issue with the .htaccess
I had not a default installation .htaccess, but a special one.
What helped in my case wa putting following into the .htaccess
    # Remove www from URL's
    RewriteCond %{HTTP_HOST} ^www\.(.+)
    RewriteRule ^ http%2://%3%{REQUEST_URI} [L,R=301]

    # Remove trailing slashes
    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteRule ^(.*)/$ /$1 [L,R=301]

But also got issues similar to this answer afterwards: https://craftcms.stackexchange.com/a/35846/11140