Random Boolean networks (RBN) have been extensively studied asmodels of genetic regulatory networks. While many studies have been devoted tothe dynamics of isolated random Boolean networks, which may considered asmodels of isolated cells, in this paper we consider a set of interacting RBNs,which may be regarded as a simplified model of a tissue or a monoclonal colony.In order to do so, we introduce a cellular automata (CA) model, where each cellsite is occupied by a RBN. The mutual influence among cells is modelled byletting the activation of some genes in a RBN be affected by that of some genes inneighbouring RBNs. It is shown that the dynamics of the CA is far from trivial.Different measures are introduced to provide indications about the overallbehaviour. In a sense which is made precise in the text, it is shown that the degreeof order of the CA is affected by the interaction strength, and that markedlydifferent behaviours are observed. We propose a classification of these behavioursinto four classes, based upon the way in which the various measures of order areaffected by the interaction strength. It is shown that the dynamical properties ofisolated RBNs affect the probability that a CA composed by those RBNs belongsto one of the four classes, and therefore also affects the probability that a higherinteraction strength leads to a greater, or a smaller, degree of order.