The rapid development of self-assembly approaches has enabled the creation of materials with desired organization of nanoscale components. However, achieving dynamic control, wherein the system can be transformed on demand into multiple entirely different states, is typically absent in atomic and molecular systems and has remained elusive in designed nanoparticle systems. Here, we demonstrate with in situ small-angle X-ray scattering that, by using DNA strands as inputs, the structure of a three-dimensional lattice of DNA-coated nanoparticles can be switched from an initial 'mother' phase into one of multiple 'daughter' phases. The introduction of different types of reprogramming DNA strands modifies the DNA shells of the nanoparticles within the superlattice, thereby shifting interparticle interactions to drive the transformation into a particular daughter phase. Moreover, we mapped quantitatively with free-energy calculations the selective reprogramming of interactions onto the observed daughter phases.