  We study the group velocities of electronic states and distributions of
currents in $\alpha-\mathcal{T}_3$ lattice ribbons under a uniform
perpendicular magnetic field. Using the effective low-energy model we analyze
all possible simple configurations of lattice termination with zigzag and
armchair boundaries. We show that the edge current depends on the type of
zigzag termination, and can be zero or finite near the edge. Also similar
dependence is observed in the case of armchair termination and is related to
the size of the ribbon. The nonzero current flowing along the edge can be used
a signature of formation of propagating edge states. Also we show the
qualitative difference in the distribution of the edge current between the case
of $\alpha=1$ (dice model) and other values of model parameter $\alpha\neq 1$
for armchair-terminated ribbons.
