We study a spatial network model with exponentially distributed link lengths on an underlying grid of points, undergoing a structural crossover from a random, Erdős-Rényi graph, to a d-dimensional lattice at the characteristic interaction range ζ. We find that, whilst far from the percolation threshold the random part of the giant component scales linearly with ζ, close to criticality it extends in space until the universal length scale ζ^{6/(6-d)}, for d<6, before crossing over to the spatial one. We demonstrate the universal behavior of the spatiotemporal scales characterizing this critical stretching phenomenon of mean-field regimes in percolation and in dynamical processes on d=2 networks, and we discuss its general implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.