  Discrete time crystals (DTC) have been demonstrated experimentally in several
different quantum systems in the past few years. Spin couplings and cavity
losses have been shown to play crucial roles for realizing DTC order in open
many-body systems out of equilibrium. Recently, it has been proposed that
eternal and transient DTC can be present with an open Floquet setup in the
thermodynamic limit and in the deep quantum regime with few qubits,
respectively. In this work, we consider the effects of spin damping and spin
dephasing on the DTC order in spin-optomechanical and open cavity systems in
which the spins can be all-to-all coupled. In the thermodynamic limit, it is
shown that the existence of dephasing can destroy the coherence of the system
and finally lead the system to its trivial steady state. Without dephasing,
eternal DTC is displayed in the weak damping regime, which may be destroyed by
increasing the all-to-all spin coupling or the spin damping. By contrast, the
all-to-all coupling is constructive to the DTC in the moderate damping regime.
We also focus on a model which can be experimentally realized by a suspended
hexagonal boron nitride (hBN) membrane with a few spin color centers under
microwave drive and Floquet magnetic field. Signatures of transient DTC
behavior are demonstrated in both weak and moderate dissipation regimes without
spin dephasing. Relevant experimental parameters are also discussed for
realizing transient DTC order in such an hBN optomechanical system.
