By far the more usual practice is to price loans on a rollover basis. Most of the funds come in the form of time deposits with fixed maturities. The banks also receive a substantial amount of call money. The going rate for call money is pretty much tied to the overnight euro rate, which in turn is tied by active arbitrage to the U.S. Fed funds rate. The main attraction of a call deposit is liquidity.
Time deposits pay more, but a penalty is incurred if such a deposit is withdrawn before maturity.
Dollars can't disappear, but they can move from center to center. It's possible, though very unlikely, that dollars in the euromarket could dry up because the holders decided to move their deposits from banks in eurocenters to banks in New York. Thus foreign banks could face a liquidity crisis. To protect against any such risk, many have negotiated standby lines with U.S. banks. Central banks have discussed at length the question of lender of last resort to the euromarket, and have concluded that each looks after its own.