We study the skew-product semiflow induced by a family of convex and cooperative delay differential systems. Under some monotonicity assumptions, we obtain an ergodic representation for the upper Lyapunov exponent of a minimal subset. In addition, when eventually strong convexity at one point is assumed and there exist two completely strongly ordered minimal subsets K 1 ⪡ C K 2 , we show that K 1 is an attractor subset which is a copy of the base. The long-time behaviour of every trajectory strongly ordered with K 2 is then deduced. Some examples of application of the theory are shown.