ODE solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity, and, implicitly, in stochastic optimisation. In this work, we propose and study the stochastic Euler dynamics - a continuous-time Markov process that is equivalent to a linear spline interpolation of a random timestep (forward) Euler method. We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE solution that shall be approximated. We first obtain qualitative insights by studying deterministic Euler dynamics which we derive through a first order approximation to the infinitesimal generator of the stochastic Euler dynamics. Then we show convergence of the stochastic Euler dynamics to the ODE solution by studying the associated infinitesimal generators and by a novel local truncation error analysis. Next we prove stability by an immediate analysis of the random timestep Euler method and by deriving Foster-Lyapunov criteria for the stochastic Euler dynamics; the latter also yield bounds on the speed of convergence to stationarity. The paper ends with a discussion of second-order stochastic Euler dynamics and a series of numerical experiments that appear to verify our analytical results.