ProfRob's answer and Andrew Steane's answer are mutually exclusive. The former is wrong, the latter is right, and here is why.
In this case, Doppler gives us current velocity and Hubble gives us current velocity for the current distance, because comoving distance and proper distance are defined to be equal at the present time.
That's the source, given by ProfRob himself: https://en.wikipedia.org/wiki/Comoving_and_proper_distances
If the comoving distance to a galaxy is denoted $\chi$, the proper distance $d(t)$ at an arbitrary time $t$ is simply given by
$d(t)=a(t)\chi$
The proper distance $d(t)$ between two galaxies at time $t$ is just the distance that would be measured by rulers between them at that time.
For the current value of the scale factor $a(t)=1$ we have $d(t)=\chi$. This means, that Doppler equation, that gives us the relative velocity between two co-moving frames at the co-moving distance, is currently appropriate for the proper distance as well, because they are defined as equal at the present time.
See for yourself. Calculate the velocity from the Doppler equation for the current CMB redshift and compare it with the velocity given by the Hubble's Law for the current value of Hubble constant (even assuming different, inconsistent measurements) and the distance equal to $ct_0$ where $t_0$ is the current age of the universe. In his recently edited answer,
ProfRob argues, that Using Hubble's law with light travel time distance yields a velocity that is roughly $c$ multiplied by the look back time as a fraction of the age of the universe. - Not At All. It yields $c \pm 0.03\ c$. If you want ProfRob, you can multiply it by $1$, not a fraction of the age of the universe. Calculate it. In addition, he explains his miscalculation by the reciprocal of Hubble constant not being exactly equal to the universe age and concludes, that therefore this velocity makes no physical sense. Not only that explanation makes no sense, but it's based on miscalculation. The interpretation of the result of proper calculation of this velocity is here.
ProfRob's start point are the values $46.5$ billion light years and $3.2\ c$ and his whole further argumentation is based on them. I'm questioning them in my other question, that is also given below with wider description. It was deleted, so I give you this instead and I hope the original will be undeleted.
On the one hand ProfRob argues, that I can't use the Doppler, because of general relativity, superluminal expansion with $3.2\ c$ (Doppler's velocity is limited to $(-c, c)$) and the fact, that the cosmological redshift is caused by the expansion, not the movement through space. Below I'm explaining, why I use the Doppler precisely because of the expansion and not the movement through space. On the other hand he writes, that if I do choose to interpret the velocity from the Doppler shift formula, then that is the relative velocity between the gas that emitted the radiation then to us now. Then and Now are two different, cosmic epochs, separated in time roughly by the age of the universe. That doesn't change the fact, that Doppler gives us current velocity. So "general relativity" and Doppler can't be both right in this case.
The argument about then and now actually applies to the case with the accelerated movement of the emission source through the non-expanding space. At the moment of emission, there was a redshift $z_e$ due to the relative velocity $v_{relE}$ of the source and the receiver at the moment. I'm introducing a shifted wavelength at the moment of emission: $\lambda_{e}=(1+z_e)\lambda_0$. Receiver gets $\lambda_r=(1+z_r)\lambda_e$ for the Doppler eq. with $v_{relR}$, that would be the relative velocity, if the emitter was moving with the constant speed from the moment of emission. If that was the case with expanding space, ProfRob would be right, but it's not. In the expanding space, photon's wavelength changes like the photon itself was a piece of a rubber band, stretched between the emitter and the receiver. That's why its redshift gives us the current, relative velocity from the Doppler eq.
In my opinion, the source of inconsistency between general relativity (which is supposedly a basis of calculations giving $46.5\ GLy$ and $3.2\ c$ values) and the Doppler is in calling the integration-based calculated recession velocity (as well as the proper size of the universe) a part of GR. Friedmann equations, as a solution of Einstein's equations, are a part of GR. Friedmann–Lemaitre–Robertson–Walker metric is a part of GR. Explicit form of the scale factor, derived from the Friedmann equations, is a part of GR. In my opinion, what is not a part of GR, is the integration of the generic metric that gives the current, proper distance. This metric equation uses the explicit form of the scale factor derived from GR and may be valid for every single spacetime frame, but if you integrate it, you get the Expanding confusion: Time integral of the reciprocal of the scaling factor from Hubble parameter equation. It was deleted, so I give you this instead and I hope the original will be undeleted.
I'd also like to quote Davis & Lineweaver (2003) given by ProfRob.
The relationship between proper time at the emitter and proper time at the observer is thus
$\Delta t=\Delta t_0 \gamma (1+v/c)$
$\quad =\Delta t_0 \sqrt{\frac{1+v/c}{1-v/c}}$
$\quad=\Delta t_0 (1+z)$
This is identical to the GR time dilation equation. Therefore using time dilation to distinguish between GR and SR expansion is impossible.
The determinant of our time dilation relative to the emission source frame at the moment of emission is redshift.
$\frac{\Delta t}{\Delta t_0}=(1+z)$
$(1+z)$ is equal to the Doppler value $\sqrt{\frac{1+v/c}{1-v/c}}$, so we can get the velocity from it.
There is one more piece of a puzzle. It seems that even ProfRob didn't read, what he recommended: Bunn & Hogg 2009. Please, find equations $(5)$ and $(6)$, read the description and compare it with the above calculation.
These are the conclusions at the end of chapter III:
We do not expect to have convinced purists who refuse even to talk about $v_{rel}$ to change their minds. They would dismiss $v_{rel}$ as a mere "coordinate velocity," not an "actual velocity," and take no interest in it. If purists have the courage of their convictions, they would say that it makes no sense to try to attach labels such as "Doppler" or "gravitational" to the observed redshift. This position is unassailable, and we have no wish to argue against it. We do claim, however, that if you wish to try to talk about $v_{rel}$, then the definition proposed in this section is the most natural way to do so. Because this definition of $v_{rel}$ results in the Doppler formula entirely explaining the redshift, we claim that you should either be a purist and refuse to try to label the cosmological redshift, or you should label it a Doppler shift.
And here is the calculation of the time left to the of the world:
$t_v=13.77\pm0.059\ \cdot 10^9\ y$
$v=0.99999835 \pm 0.00000??\ c$
$t_c$ : time of reaching the speed of light in our co-moving frame by the place of emission of CMB photons, that reach us today
$H_0\approxeq H(t_v)\approxeq H(t_c)$ (current expansion is roughly exponential and $v$ is almost equal to $c$)
$v = H_0 \cdot d_v = H_0 \cdot c t_v$
$H_0 = \frac{1}{t_v} \frac{v}{c}$
$c = H_0 \cdot d_c = H_0 \cdot ct_c$
$t_c= \frac{1}{H_0} = t_v \frac{c}{v}$
$\Delta t = t_c - t_v = (\frac{c}{v} - 1)\ t_v =$ ;)
You may calculate the velocity given by the Hubble's law for the current value of the Hubble constant and the distance $ct_v$, and compare it with the given $v$ from the Doppler equation.
