We analyse the origin of spurious temporal oscillations in computed boundary forces for an Eulerian unfitted finite element method for the incompressible Navier–Stokes equations in time-dependent domains. We study an Eulerian cutFEM time-stepping method for moving-domain problems. It combines standard BDF schemes with an implicit ghost-penalty-stabilised extension of the solution to a neighbourhood of the physical domain. As in related Eulerian approaches, the method exhibits spurious oscillations in the pressure component of the drag on the moving boundary, which compromises reliable force prediction. We show that these spurious boundary force oscillations are linked to the lack of unconditional $L^\infty(L^2)$-stability of the pressure. 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. Based on the analysis and recent development of pointwise divergence free cutFEM, we modify the finite element spaces and ghost-penalty stabilisation, leading to a novel method that is free of these oscillations in the considered setting.