Mathematical models of dynamic processes described by systems of differential-difference equations of delay type with a linearly increasing aftereffect are considered. Such a class of systems has been investigated significantly worse than the class of systems with limited aftereffect. However, in recent times many new applications have appeared in the controlled dynamic processes described by such systems. This paper is devoted to the study of the asymptotic stability of the zero solution of homogeneous differential-difference systems with several concentrated linearly increasing delays. The theoretical basis of the study is the approach of B.S. Razumikhin, which made it possible to obtain coefficient sufficient conditions for asymptotic stability. Further analysis of the asymptotic stability of nonlinear systems with unlimitedly increasing delay can be based on an adaptation of the Lyapunov-Krasovsky approach. As an application, we consider a dynamic model of the span of a large family of UAVs over a limited tunnel, which is described by a system of differential-difference equations with concentrated constant and linearly increasing delays. The approach used in the work can be applied to the analysis of stability, including systems with distributed delay.