Everything so far has stayed within one currency and one central bank's books. Cross-border payments break that assumption immediately: a payer's bank and a payee's bank may sit in different countries, use different currencies, and have no direct account relationship with each other at all, so the message-versus-settlement distinction from the message and the movement becomes the whole story. SWIFT (the Society for Worldwide Interbank Financial Telecommunication), founded in 1973 and headquartered in Belgium, describes itself precisely: a "global member-owned cooperative" providing "secure financial messaging services" to over 11,000 institutions across more than 200 countries, and, in its own words, it is not a bank: it does not hold funds, does not hold accounts on customers' behalf, and does not itself settle a single payment. What SWIFT actually carries is a standardized message, historically formats like MT103 (a customer credit transfer instruction) and MT202 (a bank-to-bank "cover" payment), with the industry migrating toward the richer ISO 20022 standard SWIFT itself maintains, that tells the receiving bank what to do, not money itself. SWIFT is overseen by a group of central banks led by the National Bank of Belgium (reflecting its Belgian headquarters), alongside other G-10 central banks including the Federal Reserve and the European Central Bank. Read that whole paragraph as one operational fact worth holding for life: "the payment went out over SWIFT" tells you a message was sent, it tells you nothing, on its own, about whether or when the underlying value has actually moved. The actual movement happens through correspondent banking.
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What is correspondent banking?
Correspondent banking is what banks use when they need to move money in a currency or a country where they have no branch and no direct account relationship with the receiving institution, they use an intermediary bank, a correspondent, that already holds an account relationship with (or is itself) the ultimate receiving bank. The account at the heart of this relationship has two names depending purely on whose books you are reading: from the perspective of the bank that owns the foreign-currency balance held abroad, it is that bank's nostro account ("ours," held with the foreign correspondent); from the perspective of the correspondent bank holding that balance for another institution, the identical account is a vostro account ("yours," held locally on the correspondent's own books). It is one account, two names, depending on which side of the balance sheet you are standing on, never two accounts, and never money "in transit" between them.
How does a cross-border payment move, hop by hop?
Trace a concrete cross-border payment hop by hop, because this is where the "whose balance sheet expands" question earns its keep across an international boundary. Suppose a US bank, Bayfront Bank, needs to send US dollars to Godavari Bank Mumbai, but the two banks have no direct account relationship. Bayfront routes the payment through Union Gateway Bank, a US correspondent at which Godavari Bank Mumbai maintains a US-dollar account. Hop 1: Bayfront's balance sheet shrinks, its reserve balance and its own customer's deposit liability both fall by the sent amount, the identical mechanic as an ordinary domestic RTGS debit. Hop 2: Union Gateway Bank's balance sheet expands, a new asset appears (its own reserve balance rises, credited via the ordinary domestic interbank system, since Bayfront and Union Gateway do have a direct relationship or share the same domestic RTGS/Fedwire rail) matched by a new liability (Godavari's vostro account balance at Union Gateway rises by the identical amount). Hop 3: Godavari Bank Mumbai's balance sheet also expands, its asset side gains (its nostro claim on Union Gateway has just grown), matched by a new liability (the ultimate India-side recipient's rupee deposit rises, once Godavari converts the dollar amount at whatever exchange rate it applies). No physical currency ever crosses an ocean. Two correspondent-chain banks' balance sheets each expand by a matched asset-and-liability pair, and the message that told them to do so, very possibly carried by SWIFT, never itself held or moved a rupee or a dollar.
What does a cross-border payment really cost?
Two costs hide inside this chain that a customer rarely sees stated plainly. First, the explicit fee a bank charges for the wire is often the smaller piece of the true cost. Second, and larger, is the FX-spread cost: the exchange rate a bank actually applies to convert the sent currency is typically worse than the mid-market rate (the rate at which large institutions trade the currency pair with each other), and that gap (multiplied across the full amount sent) is a real cost to the sender that never appears as a labeled line item, however the transfer is marketed. Worked Example 7 computes both costs precisely, in Python, and shows exactly why a transfer advertised as "$0 fee" can still cost the sender a meaningful percentage of the amount sent.
Why do correspondents settle only the net?
Finally, a correspondent relationship rarely needs to move real reserves for every single customer transfer that crosses it, for the same reason UPI's interbank layer does not: across a full settlement day, a correspondent pair typically has flows running in both directions, customers sending dollars to India, and other customers sending dollars back to the US through the identical correspondent relationship, and only the net imbalance between those two directions ever needs to actually settle as real reserves moving between the two banks. Worked Example 7 also computes this netting efficiency directly.