  Exact dynamical equations for a generic dust matter source field in a
cosmological context are formulated with respect to a non-comoving
Newtonian-like timelike reference congruence and investigated for internal
consistency. On the basis of a lapse function $N$ (the relativistic
acceleration scalar potential) which evolves along the reference congruence
according to $\dot{N} = \alpha \Theta N$ ($\alpha = {const}$), we find that
consistency of the quasi-Newtonian dynamical equations is not attained at the
first derivative level. We then proceed to show that a self-consistent set can
be obtained by linearising the dynamical equations about a (non-comoving) FLRW
background. In this case, on properly accounting for the first-order momentum
density relating to the non-relativistic peculiar motion of the matter,
additional source terms arise in the evolution and constraint equations
describing small-amplitude energy density fluctuations that do not appear in
similar gravitational instability scenarios in the standard literature.
