  Context. Most stars from in groups which with time disperse, building the
field population of their host galaxy. In the Milky Way, open clusters have
been continuously forming in the disk up to the present time, providing it with
stars spanning a broad range of ages and masses. Observations of the details of
cluster dissolution are, however, scarce. One of the main difficulties is
obtaining a detailed characterisation of the internal cluster kinematics, which
requires very high quality proper motions. For open clusters, which are
typically loose groups with some tens to hundreds of members, there is the
additional difficulty of inferring kinematic structures from sparse and
irregular distributions of stars. Aims. Here, we aim to analyse internal
stellar kinematics of open clusters, and identify rotation, expansion or
contraction patterns. Methods. We use Gaia Early Data Release 3 (EDR3)
astrometry and Integrated Nested Laplace Approximations to perform vector-field
inference and create spatio-kinematic maps of 1237 open clusters. The sample is
composed of clusters for which individual stellar memberships were known, thus
minimising contamination from field stars in the velocity maps. Projection
effects were corrected using EDR3 data complemented with radial velocities from
Gaia Data Release 2 and other surveys. Results. We report the detection of
rotation patterns in 8 open clusters. Nine additional clusters display possible
rotation signs. We also observe 14 expanding clusters, with 15 other objects
showing possible expansion patterns. Contraction is evident in two clusters,
with one additional cluster presenting a more uncertain detection. In total, 53
clusters are found to display kinematic structures. Within these, elongated
spatial distributions suggesting tidal tails are found in 5 clusters.
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