We consider a wireless sensor network, where nodes switch between an active (on) and a sleeping (off) mode, to save energy. Their switching on/off schedules are completely non-coordinated. Their positions are distributed according to a Poisson process, and their connectivity range is larger or equal to their sensing range. The durations of active and sleeping periods are such that the number of active nodes at any particular time is so low that the network is always disconnected. Is it possible to use such a network for time-critical monitoring of an area? Such a scenario requires indeed to have bounds on the latency, which is the delay elapsed between the time at which an incoming event is sensed by some node of the network, and the time at which this information is retrieved by the data collecting sink. A positive answer is provided to this question under some assumptions discussed in the paper. More precisely, we prove that the messages sent by a sensing node reach the sink with a fixed asymptotic speed, which does not depend on the random location of the nodes, but only on the network parameters (node density, connectivity range, duration of active and sleeping periods). The results are obtained rigorously by using an extension of first passage percolation theory.