Hydrodynamic interactions play a pivotal role in the dynamical behaviour of mesoscale systems such as colloidal suspensions, yet isolating their contribution from other effects remains a key challenge. Hydrodynamic correlations within a fluid are a consequence of local momentum conservation. Hence, as is commonly believed, violation of local momentum conservation should lead to non-hydrodynamic behaviour, where long-range correlations in the fluid are absent. Here, we demonstrate that generally this is a necessary but not sufficient criterion to achieve non-hydrodynamic behaviour. The motion of a massive particle leads to density modulations within the fluid. When the mechanisms underlying the relaxation processes of such modulations are removed, the dynamical behaviour of the system becomes unphysical. We show how the density relaxation mechanisms can be reintroduced in multiparticle collision dynamics (MPC) simulations, providing a consistent description of a system without hydrodynamic interactions.