We conduct a thorough study into the feasibility of measuring large-scale correlated proper motions of galaxies with astrometric surveys. We introduce a harmonic formalism for analysing proper motions and their correlation functions on the sphere based on spin-weighted spherical harmonics, and study the statistics of the transverse velocity field induced by large-scale structure. We use a likelihood formalism to derive optimal estimators for the secular parallax due to the Solar System's motion relative to distant objects, and compute the variance and bias due to peculiar velocities and relativistic aberration. We use a simulated catalogue of galaxy proper motions with radial distributions and noise properties similar to those expected from Gaia to forecast the detectability of the proper motion dipole, whose amplitude may be considered a proxy for the Hubble constant. We find cosmic variance to be the limiting source of noise for this measurement, forecasting a detectability of 1-2σ on a single component of the local velocity, increasing to 2-4σ (equivalent to a 25%-50% measurement of the Hubble constant) if the CMB dipole is included as prior information. We conduct a thorough study into the radial dependence of the signal-to-noise, finding that most of the information comes from galaxies closer than a few hundred Mpc. We forecast that the amplitude of peculiar transverse velocities can potentially be measured with 10σ significance; such a measurement would offer a unique probe of cosmic flows and a valuable test of the cosmological model. 1 http://sci.esa.int/gaia/ 2 http://ngvla.nrao.edu/ like', i.e. ellipticals with large bulge-to-disk components. Additionally, a large population of quasars will be observed to pin down the celestial reference frame (see, e.g. Gaia Collaboration et al. 2018).There are several potential uses of such a dataset for cosmology (see Darling et al. 2018 for a review). Firstly, since the Solar System moves relative to distant objects, there is a 'secular parallax' (SP) proper motion in the opposite direction to our local velocity. This proper motion has an amplitude of roughly 80 r/1 Mpc µas yr −1 , where r is the comoving distance to the object, and has a dipolar dependence on angle, anti-aligned with the velocity vector of the Solar System with respect to the CMB rest frame 3 . With spectroscopic redshifts as a proxy for distance, the SP proper motion may 3 We will often express quantities in the CMB rest frame, the frame in which the CMB dipole vanishes (Planck Collaboration et al. 2014). We assume that the matter rest frame (the frame in which the dipole anisotropy of peculiar velocities vanishes) and the CMB rest frame coincide, i.e. we do not consider the 'tilted universe' scenario of Turner