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How are cross-border/foreign-currency payments handled in a payment run?

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Answer

FX payments require the correct currency, an exchange rate (spot or a treasury-agreed rate), awareness of correspondent-bank fees and cut-off times, and sometimes regulatory/beneficiary information. The payable is settled at the payment-date rate, creating a realized FX gain/loss versus the invoice-date rate, which is posted on clearing.
Invoice EUR 10,000 booked at 1.10 = $11,000 AP.
Paid at 1.12 = $11,200 cash.
  Dr AP 11,000  Dr FX Loss 200  Cr Bank 11,200
Real-world example A euro payment settled two weeks after invoicing records a $200 realized FX loss because the euro strengthened.

Common follow-ups: Who sets the FX rate for payments? | What are correspondent bank fees?