$L^2$ error estimates of semi- and full discretisations of wave equations with dynamic boundary conditions are studied, using bulk–surface finite elements and Runge–Kutta methods. The analysis resides on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed within the abstract framework. For problems with velocity terms, or with acoustic boundary conditions we prove a spatial convergence order which is less than two. These can also be observed in the presented numerical experiments.