We consider systems of nonautonomous nonlinear differential equations with infinite delay. We introduce Carath´eodory type conditions for the right-hand side in an equation, which permit one, on the one hand, to cover a fairly broad class of systems and, on the other hand, include the right-hand side in a compact function space and construct the so-called limiting equations. In the investigation, we use the construction of admissible spaces with fading memory, which permits one to obtain constructive results for the class of equations under study.