Google introduced a new service - "Google Assignments". You can sign up for the beta. It's available for users of "Google Suite for Education". I don't think the public beta is live yet, but you can already sign up. I don't have a relevant account, so I don't know for sure.
There's an option to implement it in LMS using LTI.
It's suppose to offer a complete assignments checking functionality, but the highlight is suppose to be the anti-plagiarism check.
Some thought based on the info available in the "Google Assignments" page:
* "Using the power of Search, originality reports compare student work against hundreds of billions of web pages and tens of millions of books." - What about checking the assignments against paywall articles, which are popular in higher education institution?
* "Students can run originality reports up to three times before submitting" - That would let student do reverse engineering to the system and know how to submit assignments that wouldn't cause red alarms.
* "Student work that has been scanned with the tool is not retained or owned by Google. We search for what’s publicly available on the web. In the future, we plan to add an option for schools to have a private repository of student submissions – that they own – so instructors can see peer-to-peer matches." - As they said, for now plagiarism between students won't be detected.