  We consider inf-sup stable mixed methods for the time-dependent
incompressible Stokes and Navier--Stokes equations, extending earlier work on
the steady (Navier-)Stokes Problem. A locking phenomenon is identified for
classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart
elements by a novel, elegant and simple numerical analysis and corresponding
numerical experiments, whenever the momentum balance is dominated by forces of
a gradient type. More precisely, a reduction of the $L^2$ convergence order for
high order methods, and even a complete stall of the $L^2$ convergence order
for lowest-order methods on preasymptotic meshes is predicted by the analysis
and practically observed. On the other hand, it is also shown that
(structure-preserving) pressure-robust mixed methods do not suffer from this
locking phenomenon, even if they are of lowest-order. A connection to
well-balanced schemes for (vectorial) hyperbolic conservation laws like the
shallow water or the compressible Euler equations is made.
