Currency trading through internet is permissible if there is no delay in completing the transaction. (Rolling Commission).

The Prophet (Sallallahu Alaihi wa Sallam) said: "Gold for gold and silver for silver should not be exchanged except the same amount by the same amount, and none of the two (gold or silver) should be sold except if available and present and provided it is from hand to hand. Reported by Bukhari and Muslim.

Cash currencies are like gold and silver in rate, so the same ruling applies to them. However, currency trading hand to hand is permissible if the exchange or receipt of the currency takes place in the same meeting.
Receiving money in a transaction is the same as having the amount of purchase debited from the bank accounts immediately.

When the bank gives statement to its customer of an exchange, this is considered as receiving the amount.
In addition to the condition of exchanging hand to hand, there is another condition if the currency is of the same kind, then it should be the same amount, like a dollar for a dollar, and a Euro for a Euro.

But if the two currencies differ in essence, then the only condition is that they just have to be exchanged hand to hand. So, the transactions you describe are permissible.
Allah knows best.