If a fluid is flowing along a vertical line with a constant velocity, will the pressure at every point be the same and irrespective of height?
The pressure must be different because the fluid is in equilibrium (moving at a constant velocity), but the force of gravity is acting on it downwards. This can only be balanced by a pressure difference:
Resolving forces vertically for a cube of fluid with cross-sectional area $A$ and height $\Delta z$: $$(p + \Delta P)A = pA + \rho g A \, \Delta z $$
The areas cancel, and so do the main pressure terms, leaving:
$$\Delta p = \rho g \Delta z$$
which is pascal's law.
In order for it to flow with constant velocity (and assuming incompressible), it must have constant area, as in a vertical pipe. So @alexdavey is right, for non-viscous fluid or slow velocity.
If the fluid is viscous, there will be wall drag. If the velocity is downward, the drag will counteract gravity, so there will be a terminal velocity. In that case, the pressure will not depend on $z$.