There is an expectation that the redshift of an object will change with time. The details depend on the cosmological parameters (a plot is shown below).
This was first explored by Sandage (1962) who predicted that, in a matter-dominated universe (i.e. no consideration of dark energy back then), the redshift should decrease due to the braking action of gravity. The typical numbers quoted for a galaxy at $z=0.4$ at the present epoch were a change in redshift of $\sim -5\times10^{-6}$ km/s per year.
Sandage also noted that measuring such shifts seemed impossible with the instruments of the day.
Science marches on: we now have dark energy to contend with and the possibility/probability that the expansion is accelerating. This could lead to an increasing redshift with time at redshifts below which dark energy dominates the expansion dynamics. So, a fascinating piece of work was done by Liske et al. (2008) who revisited this question and specifically looked at whether the effect could be detected using extremely large telescopes and new instruments. Their conclusion is simple: using the new European ELT (due to start working 2024) it will be possible to measure the effect in the Lyman alpha absorption lines seen along the line of sight towards quasars at redshifts of 2-5. However, it will need a 20 year baseline and about 400 nights of telescope time!
Here are some pictures from that paper. The first shows how the observed redshift of an object will change with cosmic time for different values of the main cosmological parameters. The x-axis is time (in Gyr, assuming the present-day Hubble constant is 70 km/s per Mpc), with zero representing now (note that different models predict a different age for the universe). The "concordance model" at the moment is that
represented by the red lines ($\Omega_M=0.3$, $\Omega_\Lambda=0.7$), where the three different lines represent quasars with different redhifts at the present day. I guess the thing you should take away from this diagram is that different models predict different gradients in these plots - i.e. different rates of change of the redshift with time.
Also note that the gradient now depends on the redshift. For the red concordance model, high redshifts are getting smaller with time, but low redshifts ($z<2$, where dark energy accelerates the expansion) are getting bigger with time.
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The paper then goes on to perform extensive simulations predicting how well a high resolution spectrograph on the E-ELT could perform with a 20 year observing baseline. The results are shown below, with predicted data points measured from the Lyman alpha absorption lines towards a set of quasars at various redshifts. The y-axis shows the predicted drift rate as a function of redshift. Appreciate the numbers! Even over 20 years, the changes in redshift amount to around 0.1 m/s, yet nevertheless, the prediction is that it can be observed and can be used as a model-independent way of directly measuring the expansion rate of the universe.
A slightly more accessible version of this work can be found in the ESO Messenger.
EDIT: There are also plans afoot to measure these effects at radio wavelengths using the Square Kilometre Array (due to commence operation in 2025?). Simulations by Klockner et al. (2015) suggest that precise measurements of the 21 cm (1.42 GHz) hydrogen line for ten million galaxies, each with a precison of around 10 m/s are required. The redshift drift would amount to a change in the line frequency at a given redshift of around 0.1 Hz (2 cm/s !) over a decade.
