  Signposts of early planet formation are ubiquitous in substructured young
discs. Dense, hot and high-pressure regions formed during gravitational
collapse process, integral to star formation, facilitate dynamical mixing of
dust within the protostellar disc. This provides an incentive to constrain the
role of gas-dust interaction and resolve zones of dust concentration during
star-disc formation. We explore if thermal and dynamical conditions developed
during disc formation can generate gas flows that efficiently mix and transport
well-coupled gas and dust components. We simulated the collapse of dusty
molecular cloud cores with the hydrodynamics code PLUTO augmented with
radiation transport and self-gravity. We used a 2D axisymmetric geometry and
follow the azimuthal component of velocity. Dust was treated as Lagrangian
particles that are subject to drag from the gas, whose motion is computed on a
Eulerian grid. We considered 1, 10 and 100 micron-sized neutral spherical dust.
Importantly, the equation of state accurately includes molecular hydrogen
dissociation. We focus on molecular cloud core masses of 1 and 3 Msun and
explore effects of initial rotation rates and cloud core sizes. Our study
underlines mechanisms for early transport of dust from inner hot disc regions
via the occurrence of meridional flows and outflow. The vortical flow fosters
dynamical mixing and retention of dust while thermal pressure driven outflow
replenishes dust in the outer disc. Young dynamical precursors to
planet-forming discs exhibit regions with complex hydrodynamical gas features
and high-temperature structures. These can play a crucial role in concentrating
dust for subsequent growth into protoplanets. Dust transport, especially, from
sub-au scales surrounding the protostar to outer relatively cooler parts,
offers an efficient pathway for thermal reprocessing during pre-stellar core
collapse. [Abridged]
