This paper concerns the numerical solution of stiff initial value problems for systems of ordinary differential equations. We focus on the class of waveform relaxation methods, which was introduced by Lelarasmee et al. (1982). In waveform relaxation methods, a so-called continuous time iteration is set up, which is based on a decoupling of a given initial value problem into a number of subsystems. The continuous time iteration generates a sequence of functions that approximate the solution to the given initial value problem. After discretization of the initial value problems in the continuous time iteration, one obtains a so-called discrete time iteration. In this paper we investigate the convergence of continuous time and discrete time iteration processes. We consider discrete time iteration processes that are obtained from Runge-Kutta methods, and derive convergence results that are relevant in applications to nonlinear, nonautonomous, stiff initial value problems.