When you use a tap-to-pay card to enter the subway, what happens?

There’s some kind of communication over NFC, but after that, how does the OMNY system talk to e.g. the Visa network? Different methods have different speeds; tap-to-pay on my Chase credit card is much slower than apple pay on my phone using the apple card.

The wikipedia articles on both don’t describe how they’re implemented and a cursory search didn’t turn up a blog post.