What is the cosmological redshift drift effect? What is the redshift drift effect in cosmology?
What are the necessary cosmological conditions for there to be a measurable redshift drift effect?
 A: Redshift drift is the name given to the effect whereby as the universe expands, the redshift of an object will change with time. i.e. It will drift.
A galaxy at a fixed co-moving distance will have a redshift that changes with time and the first and second time derivatives of the redshift may prove to be valuable probes of cosmological models (e.g. Meliá 2016). 
The size of the effect is of order 10  cm/s/yr, with the difference between different cosmological models being smaller than this. It is expected that the E-ELT telescope may be able to measure redshift drift over 5-year periods, using large ensembles of galaxies at similar redshifts.
An expression for the first order redshift drift is
$$\frac{dz}{dt_0}= (1+z)H_0 - H(z),$$
where $H_0$ is the current Hubble parameter at time $t_0$ and $H(z)$ is the value of the Hubble parameter at an epoch corresponding to a redshift $z$.
The condition that there be a redshift drift is therefore that 
$$ H(z) \neq (1+z)H_0$$
which is certainly the case for arbitrary values of $z$ in $\Lambda$CDM cosmology$^*$ where
$$H(z) = (1+z) H_0\left[ \Omega_r (1+z)^2 + \Omega_m (1+z) + \Omega_k + \Omega_{\Lambda}(1+z)^{-2}\right]^{1/2}$$
and the condition for a redshift drift is that
$$\left[ \Omega_r (1+z)^2 + \Omega_m (1+z) + \Omega_k + \Omega_{\Lambda}(1+z)^{-2}\right] \neq 1$$
$^*$ Other cosmologies are possible...
