Selective optical excitation of a substrate lattice can drive phase changes across heterointerfaces. This phenomenon is a nonequilibrium analogue of static strain control in heterostructures and may lead to new applications in optically controlled phase change devices. Here, we make use of time-resolved nonresonant and resonant x-ray diffraction to clarify the underlying physics and to separate different microscopic degrees of freedom in space and time. We measure the dynamics of the lattice and that of the charge disproportionation in NdNiO3, when an insulator-metal transition is driven by coherent lattice distortions in the LaAlO3 substrate. We find that charge redistribution propagates at supersonic speeds from the interface into the NdNiO3 film, followed by a sonic lattice wave. When combined with measurements of magnetic disordering and of the metal-insulator transition, these results establish a hierarchy of events for ultrafast control at complex-oxide heterointerfaces.