This presentation focuses on two aspects related to the physics of unmagnetized, relativistic collisionless shocks, mediated by Weibel-type instabilities. Its first part concentrates on the interplay of the self-induced electromagnetic turbulence and background plasma in the precursor of the shock. In such environments, there exists a non-inertial frame where the turbulence is quasi magnetostatic. We derive an analytical model describing the diffusion of the background plasma particles in this frame and, using a Monte Carlo solver, we describe its heating and slowing down along the shock precursor. In the second part, we address the nonlinear evolution of the current filaments generated by the Weibel instability. Using the Floquet theory, we perform a linear stability analysis of a periodic system of relativistic current filaments described by a relativistic fluid model. Solving for the full set of eigenmodes, we show that the two-stream plasma is susceptible to coalescence- and kink-type instabilities, and identify the regimes of dominance for each as a function of the level of nonlinearity and asymmetry of the system. All our theoretical predictions are supported by particle-in-cell simulations.