This issue is thrown by the mod_security module being activated on your server. WP Staging is firing multiple subsequent ajax post requests to admin-ajax.php to copy files on your server from one location to another. mod_security is blocking these requests after a while because it thinks that these requests are malicious ones. They are not of course.
- Increase setting: File Copy Limit to 250
- Disable mod_security at all (not recommended)
- Ask your hoster if they can provide you with the mod_security log which you send to us or just ask them if there is any specific rule which is causing mod_security to block these ajax requests.