  The Standard Cosmological Model assumes that more than 85\% of matter is in
the form of collisionless and pressureless dark matter. Unstable decaying dark
matter has been proposed in the literature as an extension to the standard cold
dark matter model. In this paper we investigate a scenario when dark matter
decays and the resultant particle moves with respect to the dark matter. A
covariant hydrodynamical model is developed in which the decay is modeled by
the transfer of energy-momentum between two dark dust fluid components. We
parameterise the model in terms of the decay rate $\Gamma$ and injection
velocity $v_i$ of the resultant dark matter particles. We apply the framework
to study the evolution of cosmic voids which are environments with low content
of baryonic matter. Thus, unlike baryon-rich environments, voids provide an
opportunity to measure dark matter signals that are less contaminated by
complex baryonic processes. We find that the growth of S-type voids is modified
by the dark matter decay, leading to imprints at the present day. This paper
serves as a proof-of-concept that cosmic voids can be used to study dark mater
physics. We argue that future cosmological observations of voids should focus
on signs of reported features to either confirm or rule out the decaying dark
matter scenario. Lack of presence of reported features could put constraints of
the decay of dark matter in terms of $\Gamma > H_0^{-1}$ and $v_i<10$ km/s.
