We consider the cause and cure of spurious temporal oscillations of boundary forces in the context of an unfitted finite Element for the incompressible Navier–Stokes equations in time-dependent domains. We focus on an Eulerian method for moving domain problems based on cut-FEM. This method enables standard BDF time-stepping by an implicit discrete extension of the solution using ghost-penalty stabilisation to a neighbourhood of the physical domain. We demonstrate that the presence of spurious boundary forces can be explained mathematically by the lack of unconditional stability of the discrete pressure in the $L^\infty(L^2)$-norm. This result is then related to the previous analysis of the moving domain method for the Stokes and linearised Navier-Stokes equations in the literature. We further investigate the dependence of spurious oscillations observed in the Eulerian finite element method on a number of model and discretisation parameters. These results show that the approach is competitive to other approaches to reduce spurious pressure oscillations. Finally, we propose a modification of these methods that is oscillation-free based on the previous analysis.