Luminescent defects in hexagonal boron nitride (h-BN) have recently emerged as a promising platform for non-classical light emission. On-chip solutions, however, require techniques for controllable in-situ manipulation of quantum light. Here, we demonstrate the dynamic spectral and temporal tuning of the optical emission from h-BN via moving acoustomechanical modulation induced by stimulated phonons. When perturbed by the propagating acoustic phonon, the optically probed radiative h-BN defects are periodically strained and their sharp emission lines are modulated by the deformation potential coupling. This results in an acoustically driven spectral tuning within a 2.5-meV bandwidth. Our findings, supported by first-principles theoretical calculations, reveal exceptionally high elasto-optic coupling in h-BN of~50 meV/%. Temporal control of the emitted photons is achieved by combining the acoustically mediated fine-spectral tuning with spectral detection filtering. This study opens the door to the use of sound for scalable integration of h-BN emitters in nanophotonic and quantum information technologies.