Uncertainties in the shape of a moving domain can have a significant impact on the solution of partial differential equations posed on it, including models arising in fluid mechanics and problems motivated by fluid–structure interaction. Accounting for such geometric uncertainties is therefore essential for reliable simulations, but it leads to substantial computational costs. In this talk, we address this challenge by integrating the treatment of uncertain geometries with an arbitrary Lagrangian–Eulerian (ALE) framework for domain motion, and using multilevel Monte Carlo sampling for the efficient estimation of statistical quantities of interest. We present convergence and complexity analyses for elliptic and saddle-point problems, focusing on the Poisson equation and the steady-state Stokes equation on random, time-dependent domains. These results are supported by numerical experiments on standard benchmark problems, where we observe computational savings of up to an order of magnitude compared to classical Monte Carlo sampling. We conclude with a biomechanically motivated example illustrating the practical impact of the proposed strategy.