Proper motion studies of stars in the centre of the Milky Way have typically been limited to the Arches and Quintuplet clusters, and to the central parsec. Here we present the first results of a large-scale proper motion study of stars within several tens of parsecs of Sagittarius A* based on our 0.2″ angular resolution GALACTICNUCLEUS survey (epoch 2015) combined with NICMOS/HST data from the Paschen-α survey (epoch 2008). This comprises the first extensive proper motion study of the central ∼36′×16′ of the Galaxy, which is not covered adequately by any of the existing astronomical surveys, such as Gaia, because of the extreme interstellar extinction (A_V ≳ 30 mag). Proper motions can help us to disentangle the different stellar populations along the line-of-sight and interpret their properties in combination with multi-wavelength photometry from GALACTICNUCLEUS and other sources. It also allows us to infer the dynamics and interrelationships between different stellar components (Galactic bulge, nuclear stellar disk, nuclear stellar cluster) of the Galactic centre (GC). In particular, we use proper motions to detect co-moving groups of stars which are able to trace low-mass or partially-dissolved young clusters in the GC that can hardly be discovered by any other means. Our pilot study for this work is based on a field in the nuclear bulge associated with HII regions that show the presence of young stars. We have detected the first group of co-moving stars coincident with an H II region. Using colour–magnitude diagrams, we have inferred that the co-moving stars are consistent with the post-main sequence stars with ages of few Myr. Simulations show that this group of stars is a real group that can indicate the existence of a dissolving or low-to-intermediate-mass young cluster. A census of these undiscovered clusters will ultimately help us to constrain star formation at the GC in the past few ten Myr.