The network will soon start to reject some of the older transactions and the transactions with the higher fees will get confirmed, the situation will fix itself soon. When there are thousands of backlogged transactions, the average transaction fee goes up which gives miners an incentive to keep mining, I'm certain that the miners will continue to mine. If they stop for some reason, i guess that the network will reject the transactions and the coins would be returned to the wallet that they're being sent from.