Does Deep Conversions work if my checkout is on Stripe Checkout or another payment provider domain?

Yes. That is exactly what it is built for. Deep Conversions bridges external and third-party payment pages without dropping cookies. The AI analyzes the checkout page for high-confidence payment actions, the server persists the signed journey across the domain change, and the Stripe webhook confirmation closes the loop with a server-side CAPI Purchase event attributed to the original ad click.