  We study a spatial network model with exponentially distributed link-lengths
on an underlying grid of points, undergoing a structural crossover from a
random, Erd\H{o}s--R\'enyi 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
this {\em 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.
