A new class of time-dependent background space configurations is shown to be exact classical string vacua. These vacua have variations in the compactified internal space propagating through the uncompactified space at the speed of light. All the moduli in toroidal compactification can depend on time in this way and still give a consistent compactification. The question of whether variations in the moduli of more complicated static vacua can propagate is also examined. The spaces generated in this way are often explicit string vacua, without readjustment at higher orders in alpha'.