  In this paper, we investigate the dynamics behaviors of genuine multipartite
Einstein-Podolsky-Rosen steering (GMS) and genuine multipartite nonlocality
(GMN), and explore how to recover the lost GMS and GMN under a mixed
decoherence system. Explicitly, the decoherence system can be modeled by that a
tripartite Werner-type state suffers from the non-Markovian regimes and one
subsystem of the tripartite is under a non-inertial frame. The conditions for
steerable and nonlocal states can be obtained with respect to the tripartite
Werner-type state established initially. GMS and GMN are very fragile and
vulnerable under the influence of the collective decoherence. GMS and GMN will
vanish with growing intensity of the Unruh effect and the non-Markovian
reservoir. Besides, all achievable GMN's states are steerable, while not every
steerable state (GMS's state) can achieve nonlocality. It means that the
steering-nonlocality hierarchy is still tenable and GMN's states are a strict
subset of the GMS's states in such a scenario. Subsequently, we put forward an
available methodology to recover the damaged GMS and GMN. It turns out that the
lost GMS and GMN can be effectively restored, and the ability of GMS and GMN to
suppress the collective decoherence can be enhanced.
