General criteria are given to ensure that in a family of discrete random processes, given parameters exhibit convergence to the solution of a system of differential equations. As one application we consider random graph processes in which the maximum degree is bounded and show that the numbers of vertices of given degree exhibit this convergence as the total number of vertices tends to infinity. Two other applications are to random processes which generate independent sets of vertices in random $r$-regular graphs. In these cases, we deduce almost sure lower bounds on the size of independent sets of vertices in random $r$-regular graphs.