We study the convergence in distribution of the renormalized error arising from the discretization of a Brownian semimartingale sampled at stopping times. Our mild assumptions on the form of stopping times allow the time grid to be a combination of hitting times of stochastic domains and of Poisson-like random times. Remarkably, a Functional Central Limit Theorem holds under great generality on the semimartingale and on the form of stopping times. Furthermore, the asymptotic characteristics are quite explicit. Along the derivation of such results, we also establish some key estimates related to approximations and sensitivities of hitting time/position with respect to model and domain perturbations.