This paper studies the propagation and crossing of signals in cellular automata whose cells are updated at random times. The signals considered consist of a core part, surrounded by an insulating sheath that is missing at the side of the core that corresponds to the direction into which the signal moves. We study two types of signals: (1) signals by which the sheath at the left and right sides of the core advance first in a propagation step, followed by the core, and (2) signals by which the core advances first, followed by the sheath at its left and right sides. These types naturally arise in, respectively, Moore neighborhood cellular automata with semi-totalistic rules and von Neumann neighborhood cellular automata with symmetric transition rules. The type of a signal has a profound impact on the way signals cross each other, as we show by the construction of one signal of each type. The results we obtained should be of assistance in constructing asynchronous circuits on asynchronous cellular automata.