They would have to do the same mining process again as they have built an entirely new block by including a single different transaction.
So lets say in the Bitcoin network (I guess you are more familiar with that than with LTC) there hasn't been any block minted in 30 minutes, the current block is 468359 and everyone is mining block 468360. They have all been mining for 30 minutes but then I broadcast a high fee transaction (900 satoshi/Byte) and my transaction gets included in block 468360. 1. How does that work? 2. After I broadcast my transaction do they all start their mining over?