I'm not an expert on this part of the compiler, but what I can tell you is that Rust uses multiple forms of IR, and the IR that the borrow checker operates on (
https://rustc-dev-guide.rust-lang.org/mir/index.html) already encodes control flow. So doing that construction, at least, is just a normal part of the compiler, and isn't part of the borrow checker itself.
However, in my understanding, it takes that IR, does build a borrow graph, computes liveliness, and then does inference. There's been a number of details in how this has changed over the years, but it's vaguely datalog shaped.