The recent literature on time series has developed a lot of models for the analysis of the dynamic conditional correlation, involving the same variable observed in different locations; very often, in this framework, the consideration of the spatial interactions are omitted. We propose to extend a time-varying conditional correlation model (following an ARMA dynamics) to include the spatial effects, with a specification depending on the local spatial interactions. The spatial part is based on a fixed symmetric weight matrix, called Gaussian Kernel Matrix (GKM), but its effect will vary along the time depending on the degree of time correlation in a certain period. We show the theoretical aspects, with the support of simulation experiments, and apply this methodology to two space-time data sets, in a demographic and a financial framework respectively.