This paper considers robotic sensor networks performing spatially-distributed estimation tasks. A robotic sensor network is deployed in an environment of interest, and takes successive point measurements of a dynamic physical process modeled as a spatio-temporal random field. Taking a Bayesian perspective on the Kriging interpolation technique from geostatistics, we design the DISTRIBUTED KRIGED KALMAN FILTER for predictive inference of the random field and of its gradient. The proposed algorithm makes use of a novel distributed strategy to compute weighted least squares estimates when measurements are spatially correlated. This strategy results from the combination of the Jacobi overrelaxation method with dynamic average consensus algorithms. As an application of the proposed algorithm, we design a gradient ascent cooperative strategy and analyze its convergence properties in the absence of measurement errors via stochastic Lyapunov functions. We illustrate our results in simulation.