Note that this is done for “existing formally verified C codebases” which is a lot different from typical systems C code which is not formally verified.
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The concept of the borrow checker has been on a simplified version of rust https://people.mpi-sws.org/~dreyer/papers/rustbelt/paper.pdf - work has continued in this area steadily (e.g. see tree borrows)
There's a variety of tools that take rust code and translate it to something a proof system understands, and then checks that it matches a specification. AWS is leading a project to use these to verify the standard library: https://aws.amazon.com/blogs/opensource/verify-the-safety-of...