Motor-driven cytoskeletal remodeling in cellular systems can often be accompanied by a diffusive-like effect at local scales, but distinguishing the contributions of the ordering process, such as active contraction of a network, from this active diffusion is difficult to achieve. Using light-dimerizable kinesin motors to spatially control the formation and contraction of a microtubule network, we deliberately photobleach a grid pattern onto the filament network serving as a transient and dynamic coordinate system to observe the deformation and translation of the remaining fluorescent squares of microtubules. We find that the network contracts at a rate set by motor speed but is accompanied by a diffusive-like spread throughout the bulk of the contracting network with effective diffusion constant two orders of magnitude lower than that for a freely-diffusing microtubule. We further find that on micron scales, the diffusive timescale is only a factor of approximately 3 slower than that of advection regardless of conditions, showing that the global contraction and long-time relaxation from this diffusive behavior are both motor-driven but exhibit local competition within the network bulk.