The evolution of density profiles in current-carrying inhomogeneous plasma is examined for evolution rates that are slow in comparison with acoustic and Alfven velocities. When the density profile is smooth in comparison with $l_{\mathrm{T}}=T/Ee$, diffusion is initially unimportant and evolution can often be described by the equation for a simple nonlinear wave. The breaking of this wave leads to the appearance of regions with a steep density gradient, i.e., diffusion shocks. The exception is ambipolar diffusion in simple plasma consisting of ions and electrons of a single kind and having constant mobilities. Examples of such stationary and moving shocks in ionospheric, gasdischarge, and semiconductor plasmas, and in electrolytes, are discussed.