We study the survival probability of moving relativistic unstable particles with definite momentum $\vec{p} \neq 0$. The amplitude of the survival probability of these particles is calculated using its integral representation. We found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-expo\-nen\-tial form of the survival probability, should occur much earlier than it follows from the classical standard approach resolving itself into replacing time $t$ by $t/\gamma$ (where $\gamma$ is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities should tend to zero as $t\rightarrow \infty$ much slower than one would expect using classical time dilation relation. Here, we show also that for some physically admissible models of unstable states, the computed decay curves of the moving particles have a fluctuating form at relatively short times including times of the order of the lifetime.