  The first part of the paper is devoted to studying the continuous dependence
of the solutions of Carath\'eodory constant delay differential equations where
the vector fields satisfy classical cooperative conditions. As a consequence,
when the set of considered vector fields is invariant with respect to the
time-translation map, the continuity of the respective induced skew-product
semiflows is obtained. These results are important for the study of the
long-term behavior of the trajectories. In particular, the construction of
semicontinuous semiequilibria and equilibria is extended to the context of
ordinary and delay Carath\'eodory differential equations. Under appropriate
assumptions of sublinearity, the existence of a unique continuous equilibrium,
whose graph coincides with the pullback attractor for the evolution processes,
is shown. The conditions under which such a solution is the forward attractor
of the considered problem are outlined. Two examples of application of the
developed tools are also provided.
