I think we all can agree, that no market moves on it's own. Movements happen in all markets and similiar moves are often seen in related markets. In indices you see them, because the Buy/Sell-programms affect not one but multiple markets.
In the currency markets it's happening, because we don't trade currencies per se, we always trade the relationship of 2 currencies. The exchange rate tells you the worth of one currency expressed in another currency.
Now let's look at the 2 major pairs. Euro/USD and USD/Yen. If the Euro/USD moves it shouldn't affect the USD/Yen and vice versa, because we have two seperate markets.
But there is alse the Euro/Yen market.
This market must move if the Euro/USD exchange rate changes and the USD/Yen market stays fixed. Why?
Because you can either trade the Euro/USD directly or you trade Euro/Yen and USD/Yen. Meaning, you can go long Euro, by going straight long Euro/USD, or You go long Euro/Yen and you go short USD/Yen.
E/U = 1.4031
E/Y = 164.47
U/Y = 117.22
If no arbitrage exists
1.4031 should be 164.47 / 117.22
164.47 / 117.22 is 1.403088 so when trading sufficcent quantities there is currently a slight arbitrage available, but the Bid/Ask spread will take care, that it is nil, the moment you try to exploit it during normal market operations.
Now say the E/U rate goes down from 1.4031 to 1.4000.
If U/Y stays fixed at 117.22 as the european and the japanese markets are not correlated and the E/Y rate stays fixed a huge arbitrage possibility would be available to any trader fast enough to grab it. Be sure there are programs out in the market trying to do just that and they act with lightning speed. They will grab any arbitrage chance opening up. So it is 100% sure that E/Y will move if the E/U moves and U/Y stays fixed.
But why would you go long Euro/USD by trading E/Y and U/Y. Normally you won't. Unless you have already a position in one of these markets, there is no compelling reason.
But if you want to go long E/Y and you know this market can not provide the volume you need for your operation, you might as well execute the trade in the major pairs E/U and U/Y. To be long E/Y you go long E/U and you go long U/Y.
(It's long U/Y because that pair is traded inverse to E/U in the Forex market. Be aware that in the currency futures market all currency futures are traded against the USD. So there it's E/U and Y/U or Long 6E Z7 and Short 6J Z7)
Now you might notice, that E/Y or to an even greater extend the British Pound against the Yen (G/Y) can move very fast and a considerable number of pips.
So you might decide to trade that market. But trading that market without looking at the E/U and U/Y markets would be like driving in a car without right or left side-window. You need to see the cars approaching from the right or left to act accordingly. They affect the way the currency you trade directly.
Different indices affect each other, but just to an extend. If the EuroStoxx moves up, its no given that the DAX or the FTSE will move up as well or to the same extend.
But in the currencies, it's a clear mathematical relationship. If E/U moves and U/Y moves I can tell you the value of E/Y to the tens of a penny. And be sure that's exactly where E/Y will be trading.