High-density carbon nanotube networks (CNNs) continue to attract interest as active elements in nanoelectronic devices, nanoelectromechanical systems (NEMS) and multifunctional nanocomposites. The interplay between the network nanostructure and its properties is crucial, yet current understanding remains limited to the passive response. Here, we employ a novel superstructure consisting of millimeter-long vertically aligned single walled carbon nanotubes (SWCNTs) sandwiched between polydimethylsiloxane (PDMS) layers to quantify the effect of two classes of mechanical stimuli, film densification and stretching, on the electronic and thermal transport across the network. The network deforms easily with an increase in the electrical and thermal conductivities, suggestive of a floppy yet highly reconfigurable network. Insight from atomistically informed coarse-grained simulations uncover an interplay between the extent of lateral assembly of the bundles, modulated by surface zipping/unzipping, and the elastic energy associated with the bent conformations of the nanotubes/bundles. During densification, the network becomes highly interconnected yet we observe a modest increase in bundling primarily due to the reduced spacing between the SWCNTs. The stretching, on the other hand, is characterized by an initial debundling regime as the strain accommodation occurs via unzipping of the branched interconnects, followed by rapid rebundling as the strain transfers to the increasingly aligned bundles. In both cases, the increase in the electrical and thermal conductivity is primarily due to the increase in bundle size; the changes in network connectivity have a minor effect on the transport. Our results have broad implications for filamentous networks of inorganic nanoassemblies composed of interacting tubes, wires and ribbons/belts.