  We study the mobility of small-amplitude solitons attached to moving defects
which drag the solitons across a two-dimensional (2D) discrete
nonlinear-Schr\"{o}dinger (DNLS) lattice. Findings are compared to the
situation when a free small-amplitude 2D discrete soliton is kicked in the
uniform lattice. In agreement with previously known results, after a period of
transient motion the free soliton transforms into a localized mode pinned by
the Peierls-Nabarro potential, irrespective of the initial velocity. However,
the soliton attached to the moving defect can be dragged over an indefinitely
long distance (including routes with abrupt turns and circular trajectories)
virtually without losses, provided that the dragging velocity is smaller than a
certain critical value. Collisions between solitons dragged by two defects in
opposite directions are studied too. If the velocity is small enough, the
collision leads to a spontaneous symmetry breaking, featuring fusion of two
solitons into a single one, which remains attached to either of the two
defects.
