We studied the nonequilibrium short-time dynamics of a spin-$1/2$ chain with Dzyaloshinskii-Moriya interactions after a sudden quench by a transverse field. We found that inhomogeneous spin spirals with opposite chiralities propagate from the edges toward the center of the chain. This propagation is accompanied by quantum spin oscillations which decay asymptotically with time. A theoretical description of this phenomenon is given to a good accuracy with the help of numerical calculations with the adaptive time-dependent density matrix renormalization group algorithm.