No. At least in the simplest (homogeneous & isotropic)
cosmological models, there is a "cosmic time". To the extent that
galaxies can be regarded as test particles which just "go with the flow"
(i.e., neglecting the proper motion you mention later), all have the
same co-ordinate time and the same proper time.
>Their light clearly is red-shifted implying motion, but
> that motion is interpreted as the change in the fabric of space.
> Relative to local space, those galaxies have only a small proper motion.
This is correct. While we often refer (especially in popular
discussions) to the cosmological redshift as a Doppler shift, it is
actually understood in general relativity as something that happens
while the light signals are _en route_ & is not directly due to motion
of source or observer.
The difference between the 2 ideas can be seen most clearly by
considering a model in which the universe is "at rest" at the time of
emission, then expands while the light is in transit but slows down &
stops before observation. A cosmological redshift would still take
place, even though source/observer were at rest with respect to one
another at the times of emission/absorption.
This redshift is actually due to a "stretching" of waves in
proportion to the scale factor of the universe, something that can be
worked out by solving the wave equation in an expanding space.
Schroedinger's _Expanding Universes_ discusses some of these points.
When we say that a galaxy or QSO with a certain redshift is
receding from us at some fraction of c, that is really an inferred
speed, which we should think of as that galaxy acquiring (relative to
us) during the time in which the light signal is in transit.