Colliding plasmas are ubiquitous in astrophysical environments and allow conversion of kinetic energy into heat and, most importantly, the acceleration of particles to extremely high energies to form the cosmic ray spectrum. In collisionless astrophysical plasmas, kinetic plasma processes govern the interaction and particle acceleration processes, including shock formation, self-generation of magnetic fields by kinetic plasma instabilities, and magnetic field compression and reconnection. How each of these contribute to the observed spectra of cosmic rays is not fully understood, in particular both shock acceleration processes and magnetic reconnection have been proposed. We will review recent results of laboratory astrophysics experiments conducted at high-power, inertial-fusion-class laser facilities, which have uncovered significant results relevant to these processes. Recent experiments have now observed the long-sought Weibel instability between two interpenetrating high temperature plasma plumes, which has been proposed to generate the magnetic field necessary for shock formation in unmagnetized regimes. Secondly, magnetic reconnection has been studied in systems of colliding plasmas using either self-generated magnetic fields or externally applied magnetic fields, and show extremely fast reconnection rates, indicating fast destruction of magnetic energy and further possibilities to accelerate particles. Finally, we highlight kinetic plasma simulations, which have proven to be essential tools in the design and interpretation of these experiments.