Is it permissible, in Islamic law, to buy $400 from a bank, then deposit the money into an account to transfer it to another country where it will be exchanged for a different currency (such as the British Pound in the UK)?

The exchange of currencies requires actual or constructive possession. Depositing into an account and withdrawing from it, bank transfers, and receiving a valid check are considered constructive possession. It is permissible to exchange currencies of different types, even if the two عوضان (substitutes) differ in value, provided there is actual possession, or possession of the check or the remittance order. If the remittance is in a different currency, the exchange operation is performed first, then the remittance. There is no harm in withdrawing money in another currency from the same account via an ATM card.