  Using black brane solutions in 5d Lifshitz-like backgrounds with arbitrary
dynamical exponent $\nu$, we construct the Vaidya geometry, asymptoting to the
Lifshitz-like spacetime, which represents a thin shell infalling at the speed
of light. We apply the new Lifshitz-Vaidya background to study the
thermalization process of the quark-gluon plasma via the thin shell approach
previously successfully used in several backgrounds. We find that the
thermalization depends on the chosen direction because of the spatial
anisotropy. The plasma thermalizes thus faster in the transversal direction
than in the longitudinal one. To probe the system described by the
Lifshitz-like backgrounds, we also calculate the holographic entanglement
entropy for the subsystems delineated along both transversal and longitudinal
directions. We show that the entropy has some universality in the behavior for
both subsystems. At the same time, we find that certain characteristics
strongly depend on the critical exponent $\nu$.
