In the present paper, we study motions of time-delay systems that have limiting behavior for an unbounded increase in time in the case when the limit sets might not be invariant with respect to initial differential-difference equations. The concept of an asymptotic quiescent position for the trajectories of time-delay systems is introduced. By the use of the Lyapunov functionals method, sufficient conditions for the existence of an asymptotic quiescent position for systems of differential-difference equations were obtained. In the case when a general system has a trivial solution, new sufficient conditions for its asymptotic stability are obtained. Namely, the condition of the negativity of the time-derivative of Krasovskii functionals is weakened.