What happens each time a contract billed through Stripe renews?

A contract billed through Stripe shows each cycle in Get Paid as Stripe drafts it — the invoice, its hosted link, and the payment once it lands — under the same number series as everything else, so a renewal is never something you have to go and look for in Stripe. If the customer's card is declined on a cycle, that invoice says so in words and the contract reads past due on Contracts until that invoice is paid, then returns to active on its own. Nothing is charged twice: the money you see is the money Stripe settled.