I have just pushed to WPEngine and it added some constants to wp-config.php that I do not think it should have.
It changed WP_SITE_URL and WP_HOME both to the URL of the LocalWP local site, causing redirects. It does not seem to have changed any other configuration variables.
I was under the impression that Local would not make changes to wp-config.php on the WPEngine site for obvious reasons?