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 2D lattice at the characteristic interaction range $\zeta$. We find that, whilst far from the percolation threshold the random part of the incipient cluster scales linearly with $\zeta$, close to criticality it extends in space until the universal length scale $\zeta^{3/2}$ before crossing over to the spatial one. We demonstrate the universal character of the spatio-temporal length scales characterizing this critical stretching phenomenon in percolation and in dynamical processes, and we discuss its implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.