  We address the transverse mobility of spatial solitons in competing
parity-time-symmetric linear and nonlinear lattices. The competition between
out-of-phase linear and nonlinear lattices results in a drastic mobility
enhancement within a range of soliton energies. We show that within such range,
the addition of even a small imaginary part in the linear potential makes
soliton mobility strongly asymmetric. The minimal phase tilt required for
setting solitons into radiationless motion across the lattice in the direction
opposite to that of the internal current drops to nearly zero, while the
minimal phase tilt required for motion in the opposite direction notably
increases. For a given initial phase tilt, the velocity of soliton motion grows
with an increase of the balanced gain/losses. In this regime of enhanced
mobility, tilted solitons can efficiently drag other solitons that were
initially at rest, to form moving soliton pairs.
