We use confocal microscopy, rheology and light scattering to investigate the structure and the plasticity of a composite material, formed by Pluronics F108, a commercial triblock copolymer to which small amounts of colloidal nanoparticles are added. At sufficiently high polymer concentration and temperature, the block copolymer forms micelles that are regularly arranged on a FCC lattice, while the nanoparticles are preferentially segregated in the grain boundaries between micelle crystallites. We show that both the segregation efficiency and the grain size can be tuned by varying the speed at which the sample is solidified and the nanoparticle amount. We rationalize our findings by extending classical theories for nucleation and growth of crystals to account for the presence of the nanoparticles. We moreover investigate material fatigue and plasticity by measuring microscopic rearrangements in our composite material submitted to thousands of deformation cycles. Light scattering data collected at several scattering vectors and for various deformation amplitudes can be scaled onto a single master curve, exhibiting two regimes. In the first regime, the shear-induced dynamics become increasingly slower with time ("aging"), while a stationary regime where plasticity does not evolve anymore is eventually reached. Remarkably, the cross-over between the aging and the stationary regime depends on the probed length scale, larger length scales reaching the stationary state earlier. We propose a simple physical picture to rationalize these results.