A dense gas of neutrinos and antineutrinos can undergo fast pairwise conversions near the decoupling regions of core-collapse supernovae and in compact binary neutron star mergers. The flavor dependent neutrino heating can play a role in sustaining convection inside the hot and dense matter. In this paper, we study the unexplored effect of the bulk velocity of matter on fast pairwise conversions and demonstrate that depending on the direction and the magnitude of the bulk velocity, neutrino flavor conversions could be significantly enhanced or suppressed. The bulk velocity of matter, which is usually neglected in the context of neutrino oscillations, can reach values of one-tenth of the speed of light in astrophysical environments. We find that bulk velocities much smaller than the maximum allowed velocities can substantially change the neutrino flavor conversion rate. The demonstration of possible enhancement of neutrino flavor conversion rate due to the bulk velocity of matter also raises several important issues that are of relevance in the supernova mechanism. Future studies with realistic velocity profiles could elucidate the possible implications on the phenomenology of core-collapse supernovae and neutron star mergers.